SHORT ANTENNA WITH ENHANCED RADIATION OR IMPROVED DIRECTIVITY By Chun-Ju Lin AN ABSTRACT OF A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1969 ABSTRACT SHORT ANTENNA WITH ENHANCED RADIATION OR IMPROVED DIRECTIVITY BY Chun-Ju Lin A conventional electrically short, linear antenna has small radiated power and low directivity. Consequently, its practical applications are severely restricted. The purpose of this research is to investigate the feasibility of enhancing the radiated power or improving the directivity of a short antenna by a double impedance loading technique. This technique consists of mounting the appro- priately chosen lumped impedances symmetrically along the antenna surface to implement a modification of its current distribution. The current is adjusted in such a way to achieve either enhancement of its radiated power or an improvement in the directivity of the short antenna. In the theoretical study, King's modified method is applied to develop an approximate solution for the current distribution along the doubly loaded short antenna. From this solution, input impedances and typical current distributions of antennas loaded to obtain either enhanced radiation or improved directivity are de- termined. An expression for the optimum loading impedance to achieve improved directivity is established. For the enhanced radiation case, the area under the current distribution along the Chung-Ju Lin antenna with optimum loading can be increased by a factor of four relative to than that of an unloaded antenna, and its input impedance has a significantly increased resistive component and a zero re- active component. Therefore, the radiated power is greatly enhanced compared with that of the unloaded antenna. For the improved directivity case, the current distribution has a phase reversal along the antenna and the directivity corresponding to such a current is improved significantly. In addition to the doubly loaded isolated antenna, an array of doubly loaded coupled antennas is also studied, the objective again being to achieve enhanced radiation or improved directivity. An experimental study on the doubly loaded antennas, for both the enhanced radiation and improved directivity conditions, is con- ducted to verify the theoretical results. Enhanced radiation from a. coupled short antenna is also investigated experimentally. It is shown that the experimental results are in good agreement with those of the theoretical predictions. In addition to the doubly loaded antennas, the characteristics of top-loaded antennas are also in- vestigated experimentally for various types and sizes of end loadings. SHORT ANTENNA WITH ENHANCED RADIATION OR IMPROVED DIRECTIVITY By Chun-Ju Lin A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1969 Copyright by CHUN-JU LIN 1969 ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to his major professor, Dr. K.M. Chen, for his guidance and encouragement in the course of this research and for the inspiration he has given throughout the entire stage of the author's college education. He also wishes to thank the committee member Dr. D.P. Nyquist for correcting the manuscript and giving valuable suggestions in the experimental part of this research, and to the other members, Dr. H.G. Hedges, Dr. Bong Ho and Dr. R. Hamelink, for reading the thesis. Finally, the author wishes to express a special thanks to his wife, Lee-Whei, for the encouragement and understanding that only a wife can give. The research reported in this thesis was supported by the Air Force Cambridge Research Laboratories under contract AF 19(628)- 5732, ii TABLE OF CONTENTS ACKNOWLEDGMENTS ............................................ LIST OF TABLES ............................................. LIST OF FIGURES . ........................................... INTRODUCTION ............................................... CURRENT DISTRIBUTION ON A DOUBLY LOADED SHORT ANTENNA ...... 2.1 Geometry of the Doubly Loaded Short Antenna ........... 2.2 Boundary Conditions for Calculating the Antenna Current ............................................... 2.3 Integral Equation for the Antenna Current ............. 2.4 Approximate Solution of the Integral Equation ......... 2.5 Input Impedance of Doubly Loaded Short Antenna ........ SHORT ANTENNA WITH ENHANCED RADIATION ............ .......... 3 1 Introduction .......................................... 3.2 Radiated Power and Radiation Resistance ............... 3.3 Doubly Loaded Short Antenna with Increased Input Resistance and Zero Input Reactance ................... 3.3.1 Typical Current Distribution ................... 3.3.2 Input Impedance, Radiation Resistance, and Optimum Loading Impedance ...... . ....... . ... 3.3.3 Radiated Power Compared with that of Unloaded Antenna ........ ......... ............ .. ........ 3.3.4 Radiation Pattern .................... .... ...... 3.4 Loaded Short Antenna with Increased Input Resistance and Inductive Input Reactance ....... ........ . ......... 3.4.1 Typical Current Distribution ....... . ........... 3.4.2 Input Impedance ..... ... ........................ 3. 4. 3 Radiated Power ................... ......... ... 3.5 Conventional Top-Loaded (End- Loaded) Antenna ......... 3.6 Comparison of Doubly Loaded Antenna with Unloaded Base-Tuned Antenna for Radiated Power and Efficiency .. 3.7 Bandwidth of Short Antenna with Enhanced Radiation .... SHORT ANTENNAS WITH IMPROVED DIRECTIVITY 4.1 Introduction ................................... . ..... 4.2 Radiation Field of a Short Antenna with Improved Directivity .............. . ............................ iii OOOOOOOOOOOOOOOOOOO Page ii vi 12 16 21 21 22 25 25 26 31 37 38 38 39 39 42 42 44 51 Page 4.3 Optimum Loading Impedance for Improved Directivity .... 61 4.4 Typical Current Distribution on Antenna with Optimum Loading ... ....... ... ...... . ................... 64 4.5 Input Impedance of Short Antenna with Increased Directivity ........ . .................................. 67 4.6 Radiated Power and Efficiency of Antenna with Optimum Loading .......... .......... ........................... 73 DOUBLY LOADED COUPLED SHORT ANTENNAS ....................... 75 5.1 Geometry of the Doubly Loaded Short Array ............. 75 5.2 Boundary Conditions for Calculating the Antenna Current Distributions ..... ............. . . ............ 78 5.3 Integral Equations for Antenna Current Distributions .. 78 5.4 Approximate Solutions for the Antenna Currents ........ 84 5.5 Input Impedance of the Antenna Coupled with a Doubly Loaded Parasitic Element ......... . ....... ............ 96 5.6 Radiation From Coupled Short Antenna .................. 96 5.7 Enhancement of Radiated Power ........................ . 99 5.8 Improved Directivity .................................. 105 5.9 Discussion ............................................ 108 EXPERIMENTAL STUDY OF SHORT ANTENNA WITH HIGH DIRECTIVITY OR ENHANCED RADIATION ...................................... 109 6.1 Experimental Setup .................................... 109 6.2 Doubly Loaded Short Antenna ........................... 115 6.2.1 Enhanced Radiation Case ........................ 115 6.2.2 Improved Directivity Case ...................... 118 6.3 End-Loaded Short Antennas ............................. 121 6.3.1 Current Distributions on the Antenna ........... 121 6.3.2 Input Impedances ............................... 128 6.4 Short Antenna with Double Impedance and End Loadings .. 131 6.4.1 Current Distribution ................... . ...... . 131 6.4.2 Input Impedance ............................... . 131 6.5 Doubly Loaded Coupled Antennas .................. . ..... 131 REFERENCES ................................................. 142 iv Table LIST OF TABLES Input Impedance and Radiation Resistance of Short Antenna Doubly Loaded by Reactances of Various Q. V = 1 volt, f = 200 MHz, a/AO = 0.0212, d/h 2 0.7, h/AO = 0.05, x = 145 °O .................... L Input Impedance and Radiation Resistance of Short Antenna Doubly Loaded by Reactances XL of Various VO = 1 volt, f = 200 MHz, a/Ao = 0.00212A , d/h= °0. 7, h/xO =0.075, XL = 1080 0 ........ 9 ......... Input Impedance and Radiation Resistance of Short Antenna Doubly Loaded by Reactances X of Various Q. V = 1 volt, h710 =0.1, x: Input Impedances of Driven Antenna Coupled with Loaded Parasitic Element and Phase Difference Between 112(2) and 122(2) ................................... Input Impedances of a Short Antenna End-Loaded with Rectangular Bars of Various Sizes (L(cm) x k" x 1 mm thick) ................... . ............................ Input Impedances of a Short Antenna End-Loaded with Cylindrical Bars of Various Sizes (k" in diameter) Input Impedances of a Short Antenna End-Loaded with Circular Plates (1 mm thick) of Various Diameters ..... Input Impedances of a Short Antenna End Loaded with Helixes of Various Lengths (D = 3/4") ................. Input Impedances of a Short Antenna End-Loaded with Helixes of Various Diameters (L - 37.5 cm) ............ 200 MHz, a/A = 0.005121 , d/h = 0.7, 850 O ...... 9 ........... 9 ............. Page 32 106 130 LIST OF FIGURES The Doubly Loaded Short Antenna ....................... Typical Current Distribution along Antenna with Optimum Reactance Loading to make Xin = o ............ Theoretical Antenna Current Distribution Corresponding to Enhanced Radiation Condition Compared with that of Other Loadings (h = 0.1Ao, d = 0.5h) ........ ..... ..... Theoretical Antenna Current Distribution Corresponding to Enhanced Radiation Condition Compared with that of Other Loadings (h = 0.1Ao, d = 0.7h) .................. Theoretical Antenna Current Distribution Corresponding to Enhanced Radiation Condition Compared with that of Other Loadings (h = 0.1A0, d = 0.9h) ....... . .......... Theoretical Antenna Input Impedances for Zero and Optimum (enhancement) Reactive Loadings at Fixed Positions as Functions of Antenna Length .............. Theoretical Optimum Loading Reactance for Enhanced Radiation for Antennas of Various Lengths as a Function of Loading Position (h = 0.025).o ~ 0.1AO) ............. Theoretical Optimum Loading Reactance for Enhanced Radiation for Antennas of Various Lengths as a Function of Loading Position (h = 0.125).o ~ 0.2Ao) ....... . ..... Radiation Pattern of the Short Antenna with Optimum Reactance Loading ............................ . ........ Current Distribution along Antenna with Increased Input Resistance and Inductive Input Reactance ........ The Current Distributions on an Antenna with Increased Input Resistance and Input Inductive Reactance for Various Loading Reactances ............................ Efficiencies of Optimum Reactively Loaded and Base- Tuned Antennas as Functions of Inductor Q ............. vi Page 5 26 27 28 29 30 35 36 37 38 40 45 ~. Figure 3.12 The Ratio of Powers Radiated by Optimumly Loaded Antenna and Base-Tuned Antenna as a Function of Inductor Q ............................................ Theoretical Optimum Loading Reactance for Enhanced Radiation Compared with the Loading Reactance of a Fixed Inductor (L 8 0.6760 henry) ..................... Theoretical Input Impedances for Zero and Reactive Loading (L = 0.6760 henry) as Functions of Frequency . Theoretical Radiation Patterns of Short Antennas with K = 0, 1, and 2 ................................. Theoretical Radiation Patterns of Short Antennas with K = 2.33 and 3 .... ......... . .................... Theoretical Radiation Patterns of Short Antennas with K = 10 and 46 ................................... Theoretical Radiation Pattern of Short Antennas with K ~ w ........................................... Theoretical Optimum Loading Reactance for Improved Directivity (h = 0.025).O ~ 0.1Ao) ..................... Theoretical Optimum Loading Reactance for Improved Directivity (h = 0.125).o ~ 0.2Ao) ..................... Current Distribution on an Antenna with Optimum Loading for Improved Directivity ...................... Theoretical Antenna Current Distribution Corresponding to Improved Directivity Condition Compared with that of Other Loadings (h = 0.1Ao, d - 0.5h) .......... ..... Theoretical Antenna Current Distribution Corresponding to Improved Directivity Condition Compared with that of Other Loadings (h = 0.1AO, d = 0.7h) ............... Theoretical Antenna Current Distribution Corresponding to Improved Directivity Condition Compared with that of Other Loadings (h - 0.1Ao, d = 0.9h) ...... ......... Theoretical Input Impedance for Zero and Optimum (Improved Directivity) Reactive Loadings with Various Loading Positions .............................. ....... Theoretical Input Resistance for Optimum Reactively Loaded Antennas with Various Loading Positions ........ vii Page 46 49 54 65 66 67 68 69 70 72 Page The Doubly Loaded Coupled Short Antennas .............. 76 Geometry for Calculation of Radiation Field ........... 97 Theoretical Current Distributions on Doubly Loaded Coupled Antennas Corresponding to Enhanced Radiation Condition Compared with that of Other Loadings (b = 0.01A0) .......................................... 101 Theoretical Current Distributions on Doubly Loaded Coupled Antennas Corresponding to Enhanced Radiation Condition Compared with that of Other Loadings (b = 0.0310) .......................................... 102 Theoretical Current Distributions on Doubly Loaded Coupled Antennas Corresponding to Enhanced Radiation Condition Compared with that of Other Loadings (b = 0.05x0) .......................................... 103 Theoretical Current Distributions on Doubly Loaded Coupled Antennas Corresponding to Enhanced Radiation Condition Compared with that of Other Loadings (b = 0.0810) .................... . ................ ..... 104 Theoretical Radiation Patterns of Coupled Short Antennas with K1 = 0, 1, 2 and m, under Conditions of K1 = K2, K3 = 1 and a = n (O = 0) ............................ 107 Experimental Setup ..... .. ............................. 110 The Inside View of Anechoic Chamber .................. 111 TheOutside View of Anechoic Chamber ................... 111 Structure of Monopole Antenna ......................... 113 Structure of Current Probe ........................ .... 113 Experimental and Theoretical Reactance XL as a Function of Loading Resistance RL, Q = 75 .......... 116 Experimental Antenna Current Distributions Corresponding to Enhanced Radiation Condition ....................... 117 Theoretical and Experimental Input Impedances of a Short Antenna as Functions of Loading Reactance XL for the Cases of Q = 100 and Q = m ......................... 119 viii Fig: CI‘ C)“ l‘)‘\ (7“ (“h Figure 6.9 Experimental Antenna Current Distributions Corresponding to Improved Directivity Condition ....... Various End Loadings ............................... ... Experimental Current Distributions on an Antenna End-Loaded with Rectangular Bars of Various Sizes (L x k" x 1 mm thick) ............. .......... .. ..... ... Experimental Current Distributions on an Antenna End-Loaded with Cylindrical Bars of Various Sizes (D = k”) ................. ...... ....................... Experimental Current Distributions on an Antenna End-Loaded with Circular Plates of Various Diameters Experimental Current Distributions on an Antenna with Helixes of Various Lengths (D = 3/4”) ............ Experimental Current Distributions on an Antenna End-Loaded with Helixes of Various Diameters (L = 37.5 cuO ......................................... Experimental Current Distributions on a Doubly Loaded Antenna End-Loaded by a Cylindrical Bar of 8 cm .................................................. Experimental Current Distributions on a Doubly Loaded Antenna End-Loaded by a Cylindrical Bar of 10 cm ..................... ....... ............ ......... Experimental Current Distributions on a Doubly Loaded Antenna End-Loaded by a Cylindrical Bar of 12 cm .............. ........ ...... ............ ......... Experimental Current Distributions on a Doubly Loaded Antenna End-Loaded by a Circular Plate of 8 cm diameter .. ....................... ...... .......... Experimental Current Distributions on a Doubly Loaded Antenna End-Loaded by a Circular Plate of 10 cmdiameter ......OOOIOOOOOOOOI. ...... ... ........... Experimental Current Distributions on a Doubly Loaded Antenna End-Loaded by a Circular Plate of 12 cm diameter .....OOOOOOOO... ..... O ..... O ...... O ..... Experimental Current Distribution on a Doubly Loaded Antenna End-Loaded by a Helix of 2.1 cm diameter 0000000000000000000 000000000 000000000000 0.0009 Page 120 122 123 124 125 Figure 6. 6. 23 24 Page Experimental Current Distribution on a Doubly Loaded Antenna End-Loaded by a Helix of 2.8 cm diameter ........... 0.00.... ........ .... OOOOOOOOOOOOOOO 139 Experimental Current Distributions on Doubly Loaded Coupled Antennas ........................... . .......... 141 {IA r: CHAPTER 1 INTRODUCTION It is well known that a conventional short, linear antenna has small radiated power and low directivity. Therefore, much research has been conducted on improving the directivity or enhancing the radiated power of a short antenna. By using an approximate superposition method, Harrison [ 1] determined the current distribution along a doubly reactance-loaded antenna. He demonstrated that a doubly loaded linear antenna might be tuned in such a manner that its input impedance becomes purely real and its efficiency is increased relative to that of an unloaded, base tuned antenna. In this thesis an improved method is employed to solve for the current distribution on a doubly loaded short antenna, and a more comprehensive investigation is carried out. It is indicated that the antenna doubly loaded by appropriately chosen impedances has a nearly uniform current distribution between the loading points and that its input impedance has an increased re- sistive component (by a factor of two to four relative to that of an unloaded antenna) and a reactive component which vanishes. The power radiated by the short antenna is therefore increased signif- icantly. In the improved directivity case, La Paz and Miller [2:] first attempted to determine the maximum directivity theoretically available from a linear source antenna by solving for the 1 COIIEE ..v.» .C Y. ..x corresponding optimum antenna current distribution. Later, Boukamp and De Bruijin [3 ] pointed out that arbitrarily high directivities might be achieved from a linear antenna by properly adjusting its current distribution. A similar conclusion was reached by Riblet [4 1. Although no methods were suggested by the above investigators for implementing the required optimum current distributions, their researchsSimply that various degrees of improvement in the antenna directivity may be achieved by careful adjustment of the antenna current distribution. In this thesis, the optimum current dis- tribution is implemented by utilizing a double impedance loading technique, and an expression for the optimum loading impedance is developed. By applying King's modified method[fl[6]an approximate solution for the current distribution along the doubly loaded antenna is developed in Chapter 2 in terms of the antenna dimensions, its excitation frequency, and the impedance and position of the double loading. The input 1mpedance of antenna is also established in this chapter. Based on these solutions for the current distribution and input impedance, the optimum loadings for enhanced radiation and improved directivity are investigated and the numerical results are presented in Chapters 3 and 4. An extensive study of a doubly loaded array is conducted in Chapter 5 to investigate the characteristics of doubly loaded, coupled short antennas as related to the enhanced radiation and improved directivity. An experimental study of the doubly loaded antennas (both for isolated and coupled antennas) is also performed in this research. It is observed that the experimental results closely verify the theoretical prediction. In addition to the doubly impedance loaded antennas, short linear antenna end-loaded with various sizes and shapes of loading are also experimentally studied carefully in this thesis. I; .Pu ..¢ CHAPTER 2 CURRENT DISTRIBUTION ON A DOUBLY LOADED SHORT ANTENNA 2.1 Geometry of the Doubly Loaded Short Antenna: The terminology short antenna refers to a linear antenna which is not physically small, but rather one which is electrically small as measured in wave lengths. A criterion for such an antenna may be de- 2l| a 1 where so is the free-space wave number and h is the half-length of the antenna. The geometry of the short, doubly loaded, linear antenna is as indicated in Fig. 2.1. The short cylindrical antenna is assumed to be constructed of a perfect conductor of radius a and half-length h. An ideal, harmonic voltage source of angular frequency w and poten- tial Vo excites the cylinder at its center 2 = 0 (the antenna is assumed to lie along the z-axis of cylindrical coordinates), and the two identical lumped impedances ZL are loaded symmetrically on the antenna surface at z = d and z = —d. The gaps in the cylinder at the locations of the source and the loading impedances are assumed to be of length 26. Since both the source and the loading impedances are considered to be idealized point elements, then 6 is assumed to approach zero in the subsequent mathematical analysis. It is taken that the half-length h of the cylindrical antenna is very much greater than its radius a. As a result of this thin-wire 4 12(2) 12 20 + _L V Z: O o - —T__ -—~ -—’2a 12(2) 11 Z. 2T 2 = -d d 2. -h Fig. 2.1 The Doubly Loaded Short Antenna ' 9‘1“, . ,- L 311:8 .t A... 6 assumption, and due to the rotational symmetry of the cylinder, the antenna current will flow primarily along the axial or z-direction. The dimensional restrictions and axial current approximation of h >> a 2n < __ Béi< l (or A 8 << 1) o T(z) = 212(2) .... axial antenna current allow important simplification and lead to an approximate solution for the distribution of current along the antenna. It is well known [7] that subject to these restrictions 12(2) can be assumed to be con- centrated along the axis of the cylinder when calculating the vector potential at its surface with negligible error. 2.2 Boundary Conditions for Calculating the Antenna Current: The current excited on the cylinder is symmetric about its center (2 = 0) and must vanish at either of its extremities (z = :h). A pair of boundary conditions on the antenna current may therefore be expressed as I z = I -z z(> z(> (2.2) Iz<:h> = 0 From the boundary condition that the tangential component of electric field should be continuous at the surface of the antenna, it follows E:(r = a+) = E:(r = a') (2.3) where E:(r = a+) is the induced electric field just outside of the + surface of cylinder at r = a which is maintained by the current and side «I .- 7 charge on the antenna, and E:(r = a') is the electric field just in- side its surface at r = a . 2.3 Integral Equation for the Antenna Current: Since the cylindrical antenna is assumed to be constructed of . . . i . . a perfect conductor, 1ts Internal Impedance 2 per unit length is equal zero, and the electric field inside the conductor surface at r = a' is nonvanishing only at z = 0 and z = id, i.e. ZLI (d) ———E——— for -d-0 < z < -d+6 26 a V0 Ez(z) = - 33 for -0 < z < 0 (2.4) 2 I (d) L z 20 for d-0 < z < d+6 where Iz(d) and Iz(-d) are the antenna currents at the impedance a , . elements at z = :d, and Ez(z) 18 equal to zero for every other po1nt on -h S 2 S h. The total voltage drop along the antenna is therefore -Ih Ea(z)dz = V - I (d)Z - I (-d)Z . -h z o 2 L 2 L By the symmetry condition of the antenna current, eq. (2.2), Iz(-d) = Iz(d), and the last result may be expressed as -fh Ea(z)dz = v - 2I (d)Z . -h z o z L As indicated earlier, the loading impedances and source are considered as point elements of length 20 a 0, such that . h a 22:3 -f_h Ez(z)dz vo - Iz(d)ZL - Iz(-d)ZL (2.5) r lbliu? // what 40'" gm «y «I» .d DC'CEI ‘1‘ IL VI. hi. ..C “V. .1. Rd C AJ 8 By the properties of the Dirac delta function and by equations (2.4) and (2.5), then in the limit when 26 ~ 0 a Ez(z) — -v05(z) + Iz(d)ZL[6(z-d) + 6(z+d)] (2.6) where 6(2) is the Dirac delta function. The induced field just outside the surface of the antenna is given by A 5—> 1 132(2) - (-V - M (2.7) Z Z 2 where O is the scalar potential at the surface of the cylinder main- tained by the charge on the antenna, and A = E Az(z) is the vector potential at the cylinder surface maintained by the current in the antenna (it is assumed that the current is concentrated along the cylinder axis). The Lorentz condition may be applied to relate m and A as 82 O V'A+j 8I Since T(z) = E 12(2) axially directed, then A'= 2 Az(z) has only one component Az(z) and the Lorentz condition becomes 5A 82 Z S?- +-j 59 ¢ = 0 . (2.8) ’1’) 9 Substituting O in terms of AZ, according to eq. (2.8), into eq. (2.7) results in . . 2 E;> a and Boa << 1, the Helmoltz integral for the vector potential at the antenna surface can be simplified as the line integral over an axial current distribution with negligible inaccuracy, i.e. p'o h = __ ' ' ' - s S Az(z) 4n I-h 12(2 )Ka(z,z )dz for h z h where “o = the permeability of free Space -jBOR R e Ka(z,z') = Green’s function R =~/, 2 ... distance between an observation point on the surface of antenna at z and an element of current on its axis at z'. (z-z')2 + a If the left hand side of eq. (2.15) is replaced by the Helmholtz in- tegral expression, an integral equation for Iz(z) is obtained as Ih I (z')K (z z')dz' -h z d ’ ' V = - 1%1'SecBoh{ijAz(h)(CosBoz - CosBoh) - 39 SinBo(h-|Z|) 0 Z I (d) ...};EL——-[231ne h CosB d C058 2 - CosB h(SinB lz-dl 2 o o o o o + Sinao|z+d|)]}. (2.16) where C is the characteristic impedance of free space 0 = {“0 = 120w ohms. co Kd(z,z') is the difference kernel of kernel Ka(z,z') and Ka(h,z'), i.e. IE I! it/ 7 12 Kd(z,z') = Ka(z,z') - Ka(h,z') e'jBOR 'jBoRh R 0 :FW R g f(z-z')2 + a2 ’ Rh = f(h-z')?" + a2 Eq. (2.16) is a modified form of Hallen's integral equation [8] for the antenna current 12(2), and is valid for -h S 2 S h. 2-4 Approximate Solution of the Integral Equation: It is found that the current distribution on the doubly loaded short antenna can be approximated quite accurately by obtaining an approximate solution to integral equation (2.16) according to King's modified method [5][6]. This method consists essentially of assuming the current excited on the antenna to be proportional to the vector potential difference (referred to the end of the antenna). In other words, it is assumed that the ratio of vector potential difference to antenna current is relatively constant along the cylinder. Since Az(z) - Az(h) vanishes at z = ih, then Iz(z = 1h) = 0 and the induced current satisfies the boundary condition at the end of the antenna. The application of this method to the solution of integral equation (2.16) is the subject of this section. By a peaking property of the difference kernel Kd(z,z') ' = | _ ' Kd(z,z ) Ka(z,z ) Ka(h,z ) . 6(z-z') - 6(h-z') and by applying the Helmholtz integral it is found that v5.3 13 “O h Az(z) - Az(h) = Zfi’I-h Iz(z')Kd(z,z')dz' u ~ 4—3 [12(2) - 1201)] 4n or 12(2) - Iz(h) .. J; [Az(z) - Az(h)]. Since Iz(h) = 0, then 12(2) ~ Az - A200 and the induced antenna current 12(2) is therefore taken to be of the form 12(2) = CC(CosBOz - CosBoh) + CSSInBO(h-|zl) + C [ZSinB h CosB d C038 2 - CosB h(SinB Iz-dl 1 o o o o o + SinBO|z+d|)]. (2.17) Note that in eq. (2.17) Iz(z = :h) = 0 implying that the boundary condition at the antenna extremities is automatically satisfied, and the current is symmetric as it should be. The approximate solution (2.17) is further optimized by forcing it to satisfy integral equation (2.16) at z = 0. The complex constants CS, Cc’ and C1 are evaluated by sub- stituting the approximate current distribution 12(2) of eq. (2.17) back into integral equation (2.16) as 14 Ph ._ 1 1 h - 1 I I CCj_h(CosBOz CosBoh)Kd(z,z )dz + CSI_hSInBO(h-|z |)Kd(z,z )dz h . . . + Cif-hEZSIHBOh CosBOd CosBoz' - CosBOh(SInBO|z'-d| + SinBolz'+dI]Kd(z,z')dz' V . 4 . . = -j EE'Sec80h{jvoAz(h)(CosBOz - CosBoh) - 32 SlnBO(h'|Z|) zLIz(d) 4---3r-—-[23in80h CosBod CosBOz - CosBoh(SinBOIz-d| + SinBolz+d|)]}- (2.18) The complex kernel Kd(z,z') may be expressed in terms of its real and imaginary parts, i.e. e'jBoR SLjBORh E 'R. I ' I Kdr(z,z ) + JKdi(z’z ) Kd(z,z') l ' “ — - — where Kdr(z,z ) CosBOR Rh CosBORh l l ' =-—— ' - —-S' . Kdi(z,z ) Rh SinBoRh R InBOR Since Kdr(z,z') becomes very large when 2' is near 2, it follows that the principle contribution to the part of the integral that has Kdr(z,z') as kernel comes from elements of current near 2' = 2. On the other hand, K remains small when 2 = z'. This di suggests that the principal contribution to the part of the integral that has Kdi(z,z') as kernel comes from all elements of current that are at some distance from 2. Due to this peaking property of kernel Kdr(z,z') and non-peaking prOperty of kernel Kdi(z,z'), the various integrals on the left hand side of equation (2.18) may be verified numerically to have the following approximate representation: 15 [Eh(CosBoz' - CosBOh)Kdr(z,z')dZ' ~ (008302 - COSBOh) (2°19a) f§h(005802' - CosBOh)Kdi(z,z')dz' ~ (COSBOZ - COSBOh) (2°19b) IEhSinBo(h-Iz'|)Kdr(z,z')dZ' ~ SinBo(h-|2|> (2°19C) f§h31n30(h-|z'|)Rdi(z,z')dz' ~ (CosBOz - CosBoh) (2 19d) h , g . | . l I I I-hE2SInBOhCosBOdCosBOz CosBoh(Sin80|z -d|+SInBOIz +d|)]Kdr(z,z )dz N [ZSinB h CosB d C058 2 C058 h(SinB |z-d| + SinB |z+d|)] (2.19e) O O O O 0 O h - d I - I _ ' I I I f_h[281nBOhCOSBO CosBoz CosBOh(SInBO|z d|+SInBO|z +d|)]Kdi(z,z )dz N (C038 2 - CosB h) (2.19f) o o where (2.19a), (2.19c) and (2.19e) are based on the characteristics of kernel Kdr(z,z'), and the remainders, (2.19b), (2.19d) and (2.19f), are based on numerical considerations. It can be shown numerically that these remainders are roughly proportional to the shifted cosine function (CosBOz - CosBoh), These properties suggest that equation (2.18) can be split into three parts by equating corresponding terms on the right and left hand sides as follows: Ccfh (C038 2' - CosB h)K (z z')dz'+j C Ih (C098 2' - CosB h)K (z z')dz' -h o 0 dr ’ c -h o 0 di ’ + j C fh SinB (h-Iz'|)K (z z')dz' + j C fh [ZSinB h CosB d 0058 2' s -h 0 di ’ i -h o o o . u . | v 1 - CosBoh(S1nBO|z -d| + SinBOIz +dl)]Kdi(z,z )dz = %E voAz(h)SecBOh(CosBOZ - CosBoh) (2:208) va o Q 0 CSIEhSinBO(h-|z'|)Kdr(z,z')dz' = j SecBoh SinBo(h-|z|) (2.20b) 16 h . . . g . u v v Cif_h[281n80h CosBod CosBOz - CosBoh(S1nBO|z -d| + SInBolz +dI)Kr(z,z )dz 4n = u . . d - ’ - a j E-'Sec80h[231n80h CosBo CosBoz CosBoh(SinBoIz dl + SInBo|z+d|)] o (2.20c) The approximate current distribution 12(2) automatically satisfies the boundary condition Iz(z = :h) = 0. In order to optimize the approximate solution, the unknown coefficients are evaluated by forc- ing the integral equation to be satisfied at z = 0 (which has the advantage of providing accurate input impedances). This is implemented by equating z to zero in the last results as Ccfh (C058 2' - CosB h)K (o z')dz' + jC Ih [C088 2' - CosB h)K (o z')dz' -h o 0 dr ’ ' c -h o 0 di ’ + jC jh SinB (h-Iz'I)K (0 z')dz' + jC fh [2SinB h CosB d C088 2' s -h 0 di ’ i -h o o o - ' I- - I+d I I CosBoh(SinBO|z dI + SInBolz I)]Kdi(o,z )dz 4n - Co VOAZ(h)(SeCBOh-1) (2.21a) h 2nV - v y | = . O . csf_hslneo(h-|z |)Kdr(o,z )dz J Co SecBoh SlnBoh (2 21b) h . I o I _ ' I I I CiI-hEZSIHBOh CosBod CosBOz CosBoh(S1n80|z d|+51n80|z +d|)]Kdr(o,z )dz . a = -J g zLIz(d)SecBOh SinBO(h-d). (2.21c) 0 Equations (2.21a ~ c) are solved for Cs’ Ci’ and CC in terms of the antenna dimensions, the excitation frequency, the impedance and position of the double loading, and the undetermined constant terms Iz(d) and vector potential Az(h) as j2nV _ o . . s — E-¥-- SecBoh SInBoh (2 22a) 0 sdr C l7 . 4n . Ci — -J Q T, ZLIZ(d)SecBOh Sin80(h-d) (2.22b) o 1dr V T _ 4n 0 sdi . cc - g T [voAz(h)(SecBoh-l) + 21 SecBoh SlnBoh 0 cd sdr idi . - Z I (d)SecB h S1nB (h-d)] (2.22c) . L z o o 1dr where T = fh SinB (h-|z'|)K (o z')dz' sdr -h ‘ 0 dr ’ T, = Ih [ZSinB h CosB d 0088 2' - CosB h(SinB ‘z'-d| 1dr -h o o o o o + SinBO‘z'+d|]Kdr(o,z')dz' T = Ih (C058 2' - CosB h)K (o z')dz' cd -h o o d ’ T = fh SinB (h-|z'|)K (o z')dz' zdi -h 0 di ’ _ h . . . 1 Tidi f-h[231n80h CosBod CosBoz CosBOh(SInBo|z dI . I I I + SLnBOIZ +d|)]Kdi(o,z )dz . The unknown constant quantities Iz(d) and Az(h) may be evaluated by applying the conditions I (z=d) = I (d) z z u -Oh.-I II Az(h) — 4n I-h1é(z )Ka(h,z )dz where 18 To evaluate Iz(d), C5’ C1 and CC are substituted into eq. (2.17) and, by using the condition Iz(z=d) = Iz(d), the current Iz(d) is obtained in terms of Az(h) as oT Iz(d)= QT 4n D[voAz(h)(SecBOh-1) +— 2° Sdi SecBo h SinB ooh](CosB d- CosB oh) 0 cdD l sdr 2TrVo + j FIT-2T SecBoh SinBO h SinBo (h- d) (2.. 23) o sdr l where 4n T'd' D = 1 + , 1 1 z SecB h SinB (h-d)(CosB d - CosB h) 1 Q T T, L o o o o 0 cd 1dr . 8n , 2 + J Q T, ZLSBCBOh COSBOd Sin Bo(h-d). o 1dr Having evaluated 12(d) as in (2.23), it may be substituted back into eq. (2.22) and by using eq. (2.17), and the reéltion u =__9_h I 11 . . Az(h) 4n f_h12(z )Ka(h,z )dz , Az(h) IS determined completely as T A (h) = H'oTCaVo { sdi D _4fizLTidiTsdi D 2 2C T D T 5 Q0 T 3 0 cd 2 sdr cdD 1T idrT sdr _ j ATTTidiZL D } + j “ovoTsaD 5 TierlcoTsdr 4 2C oTsdrD 2 2fiu Z T, V T , _ j o L 1a 0 { sdi D + j D } (2°24) QZT D D T Ted 3 4 1dr 1 2 sdr where T = Ih (C058 2' - CosB h)K (h,z')dz' ca -h o o a T a = IEhSinBo(h-Iz'|)Ka(h,z')dz' = fh [ZSInB h CosB d CosB 2' - COSB h(Sin5 IZ'-d| -h o o o o o + SinBolz'+d|)]Ka(h,z')dz' [LII-fr L . 9W l9 a 41'fl‘idiT aZL D = 1 --—E—(3ece h-l) + C (SecB h-l)(CosB d 2 TCd O T (T )Zg D O o idr cd 0 1 - CosB h)SecB h SinB (h-d) o o o AHZLTia + ' h-l h ' - d - J T. T D C (SecBo )SecBo 51n80(h d)(CosBo CosBoh) 1dr cd 1 o 2 . . D Sec Boh SinBoh(CosBod - CosBoh)SInBO(h-d) U n SECZB h SinB h SinZB (h-d) o o o (3 fl SecB h SinB h o o The values of Iz(d) and Az(h) are completely determined by ex- pression (2.23) and (2.24), and the evaluation of the complex coeffi- cients Cs’ Cc’ and Ci is consequently completed. By combining equations (2.17), (2 22), (2 23), and (2.24) the distribution of current excited on the short, doubly loaded antenna may finally be expressed in the form V - .2 _ ' - Iz(z) - 6O{FC5(CosBOz CosBoh) + FC481n80(h lzl) - j FC [2s1n8 h CosB d C088 2 - CosB h(SinB |z-d| 2 o o o o o + Sin80|z+d|)]} (2.25) where PC = 1 {[Tca (T D - ZLTidiTSdL_ D l Tchsdr TcdDZ sd1 5 30D1TCdTidr 3 2 D T D T Z T T L 4 idi 5 sa L ia sdi . _ ________. - + h-l + 3 30D T ) J D J 30D D T (T D3 JD4)](SECBO ) TsdiDS} 1 idr 2 1 2 idr cd 20 Z _ . 1 F02 - T. SecBoh Sin80(h-d){30D FC1(CosBod - CosBoh) 1dr 1 D + j --2—-- SinB (h-d)} 30D T o l sdr T _ idi FC3 - T___ FC2 cd D . 5 FC ‘1, 4 I‘sdr This result expresses the antenna current distribution in terms of the antenna dimension, its excitation frequency and the impedance and position of the double loading. 2.5 Input Impedance of Doubly Loaded Short Antenna: The input impedance of the antenna is defined as V o . = ——————— = + . Zin Iz(z=o) Rin J Xin From eq. (2 25), this impedance can be expressed in the form . . . -1 Zin — 60{FC5(1 - CosBOh) + FCQSInBOh-j 2FCZS1nBo(h-d)} (2.26) The input impedance of result (2 26) is expressed in terms of the antenna dimensions, the excitation frequency, and the parameter of the double loading. Expressions (2.25) and (2 26) for the antenna current distribution and Input impedance of a doubly loaded short antenna are the main results of this chapter, and they will be utilized in the subsequent chapters for the study of short antennas with en- hanced radiation and high directivity. CHAPTER 3 SHORT ANTENNA WITH ENHANCED RADIATION 3.1 Introduction: A conventional short, linear antenna has a very small input resistance and a large capacitive input reactance. For an antenna having h S 0.1 A0, the input resistance is extremely small compared with that of a longer antenna, i.e., one of near resonant length. The power radiated by the antenna is strongly dependent upon its in- put resistance, since this resistance determines the degree of match- ing between the transmission system and the antenna. The radiated power of the short antenna is therefore very small due to the mis- match between its small input resistance and the characteristic impedance of a typical transmission system which might be used to excite the antenna. It is the purpose of this research to enhance the power radiated by a short antenna by increasing the input re- sistance while simultaneously tuning its input reactance to zero. In order to enhance its radiated power, the antenna should be operated at reasonance (since zero input reactance is required). This resonance condition can be implemented by inserting a low-loss series inductor at the input terminals or by loading low-loss lumped inductors on the cylindical wire antenna. In the latter case, a high Q capacitor may also be inserted at the input terminals for tuning purposes as will be indicated in Sec. 3.4. The conventional base tuning arrangement, however, Cannot increase the input resis- tance to the antenna, and therefore it is ignored in this study except for comparison purposes. 21 22 The double impedance loading is implemented by lumped in- ductors in this study. It has been found that a purely non-dissi- pative optimum loading may be utilized at various positions along the surface of the cylindrical antenna. Of course, the terminology non-dissipative loading refers to an ideal lossless inductor which cannot be realized physically. A very low-loss or high Q inductor is mounted on the antenna to implement an optimum loading in the experimental investigation. By tuning the inductor to its optimum value at a fixed position along the antenna surface, the input reactance Xin of the antenna is eliminated and its input resis- tance is increased by a factor of the order of two to four (2 ~ 4) over its value for an unloaded antenna (maximum increase is by a factor of 4), and the radiated power of the short antenna is there- fore enhanced significantly. It has also been found that for some prcper choices of loading impedances and locations, the input impedance of a short antenna can be adjusted to have a large resis- tive component and an inductive reactive component. This case appears rather attractive since the antenna can now be tuned to reasonance at its terminals with a high-Q series capacitance which is more readily implemented than a high-Q inductor. The details of all these configurations are discussed in the following sections. 3 2 Radiated Power and Radiation Resistance: The power radiated by a short antenna can be obtained easily from the well known result [7]. 23 r _ _. r B9 - ije(R0,0) (3.1) l r B v0 Ee(RO,0) 9'1 where RO is the distance from the observation point to the center of the antenna, and superscript r indicates the radiation field. r r r . . . B and A are expressed 1n spherical coordinates. The total 6 ’ ¢ 0 time-average power radiated by the antenna is thus given by 2 2 T'TUJ R0 rad. £0 E I lAgl Sine d0 (3 2) 0 r . where A can be expressed in terms of the antenna current 12(2) 0 (given by eq. (2 24)) as -‘R r ° M'0 h e 380 =..—— ’ '_ I Ae 4n Sine I-h Iz(z ) R dz 00 e’JaoRO h jeoz'COSe =-—— ' ' I 4n R Sine f Iz(z )e dz (3‘3) 0 -h subject to the usual approximations, R = R0 - z'Cose for phase factor R = R0 for amplitude terms According to the definition of a short antenna, 802' is much smaller +jBOZ'Cose than one, therefore e can be well approximated by the leading terms of a power series expansion as +jB z'C050 e O = 1 + jBOZ'Cose -'% Bgz'ZCosze . Since the antenna current is symmetric about its center 2 = 0, then 12(2) = Iz(-z) and the second term of the power series 24 integrates to zero such that equation (3.3) becomes u 15R r 0 e 0 0 h 2 .2 ______ . . _l 2 . Ae - 4n R0 Sine f h12(2 )(1 2 802 Cos 0)dz (3.4) Since 80h << 1 for a short antenna, then the second term in the integral of eq. (3.4) is normally very small compared with the first. Thus I I z(z' )dz' >>- SI $8 2Cos 29 I Z(z' )dz' , provided only that the leff side of the inequality is not forced to approach zero by reversing the phase of the current along the antenna. Eq. (3.4) can therefore be approximated by the expression , ”0 e'jBORO h n.--....... ' I I Ae 4n R Sine I Iz(z )dz 0 -h u -J’BR _ 0 0 0 . ~ 4fiRO e $1n0 AC (3.5) where Ac = I Iz(z')dz' the area under the current distribution By substituting eq. (3.5) into eq. (3.2), the radiated power is obtained as C n - O 2 2 The radiation resistance of the antenna can be defined as 2 Prad. Rrad. = 12( =0) 2 z and therefore 2 2 n Rrad = 2 :0 lAcI2 ~ _;fi£l—— ' (3“7) 3AOIz(z=0) Iz(z=0) The results of eq. (3.6) and (3.7) are very useful relations in the following sections. 25 3.3 Doubly Loaded Short Antenna with Increased Input Resistance and Zero Input Reactance: Since the input impedance of the doubly loaded antenna is a function of the impedance and position of the loading, the dimensions of the antenna, and its excitation frequency, it is difficult to formulate an expression for the optimum loading impedance required to make Xin = 0 and simultaneously maximize Rin based on the already complicated equation (2.24). Therefore the optimum impedance loading which will increase the input resistance and simultaneously provide zero input reactance will be determined by using a high speed computer to calculate and examine the antenna current and input impedance, for given antenna dimensions, with various impedances (low-loss inductors of various Q) loaded at different locations along the cylinder. 3 3.1 Typical Current Distribution Suppose that the short antenna is loaded by an optimum non- dissipative impedance of reactance [xLlop at a fixed position along its surface such that its input reactance xin is tuned to zero. It is found from eq. (2.24) by numerical calculation that the current distribution on the antenna has the general form indicated in Fig. 3.1. The amplitude of the current is almost constant between the loading points along the antenna, and decreases to zero between the loading points and the extremities of the antenna. The phase of the current is minimum and nearly constant at all points along the antenna. 26 (112(2) xL xL 22‘ N I z=h J=-d o z=d z=h N ll Fig. 3.1. Typical current distribution along antenna with optimum reactance loading to make xin = 0. The current distributions along antennas with h = 0.1710 and d = 0.5h, 0.7h and 0.9h for the case of an optimum reactance loading are plotted in Figures 3.2 to 3.4. It is found that the area under the current distribution becomes greater as the loading points are shifted toward the extremities of the cylinder. Since the radiated power is approximately proportional to the square of the area under the current distribution, this indicates that the radiated power may be enhanced more significantly if the loading is located near the end points of the antenna (provided that the optimum impedance is non-dissipative as discussed in Section 3.5). 3.3.2 Input Impedance, Radiation Resistance, and Optimum Loading Impedance The input impedance of the doubly loaded antenna is expressed by eq. (2.26) of Chapter 2 as . . -1 zin = 60{FC5(1 - CosBOh) + FC4SInBOh - jZFCZS1n80(h-d)} . From this equation, the input impedances are calculated and plotted in Fig. 3.5. The case of both zero and optimum non-dissipative impedance loadings at positions d = 0.5h, 0.7h and 0.9h along antennas of different length and constant diameter are considered at a frequency f = 200MHz. 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U a a Dr E :U _ . a x as aw x cu ON on cc On (“‘11) I2 30 :uauodmoa anxnsiaaa u 31 results that the input resistance is increased as the loading position is shifted toward the extremities of antenna. It is also indicated that the input resistance to a short antenna with optimum reactance loading is increased by a factor of two to three relative to that of an unloaded conventional antenna. The radiation resistance and input impedance of a short antenna with a lossy optimum impedance loaded at d = 0.7h are listed in Tables 3.1, 3.2 and 3.3 for various value of loading Q. These numerical results indicate how the radiation resistance and input resistance are affected by the Q of the loading impedance. The optimum loading reactances [XLJop for antenna with different half-lengths h, constant radius a, and various loading positions d are presented in Fig. 3.6 and Fig. 3.7. It is in- dicated that the optimum loading reactance [XL1op is a decreasing function of antenna length for a fixed loading position d/h, and [ijop increases for fixed h as the loading position is moved toward the extremities of antenna. 3.3.3 Radiated Power Compared with that of Unloaded Antenna It has been demonstrated in eq. (3.6) of Sec. 3.2 that the radiated power is approximately proportional to the area beneath the current distribution along the antenna. By investigating the current distributions along antennas having both zero and optimum nOn-dissipative loadings, as in Fig. 3.2 to Fig. 3.4, the radiated power of the antenna with optimum loading can be enhanced by a factor of one to four relative to that of the unloaded antenna. This con- clusion is reached by considering the matching between the antenna 32 Table 3.1 Input Impedance and Radiation Resistance of Short Antenna Doubly Loaded by Reactances XL of Various Q. v0 = 1 volt, f = 200 MHz, a/AO = 0.00212,d/h = 0.7, h/AO = 0.05, XL Q Radiation zin = Rin + j xin Resistance Rin xin 100 3.79665 30.49565 -5.0386 200 4.45551 17.80993 -4.57154 300 4.67505 13.57864 -4.47566 400 4.78479 11.46267 -4.43894 500 4.85063 10.193 -4.42050 600 4.89452 9.34653 -4.40971 700 4.92587 8.74189 -4.40273 800 4.94938 8.28841 -4.39790 900 4.96766 7.9357 -4.39438 1000 4.98229 7.65352 -4.39171 1100 4.99425 7.42265 -4.38963 1200 5.00422 7.23026 -4.38796 1300 5.01266 7.06747 -4.38659 1400 5.01990 6.92793 -4.38546 1500 5.02616 6.807 -4.38451 1600 5.03165 6.70118 -4.38369 1700 5.03649 6.60781 -4.38298 1800 5.04079 6.5482 -4.38237 1900 5.04464 6.45056 -4.38183 2000 5.04810 6.38373 -4.38135 m 5 11391 5.11391 -4.37369 1450 Table 3.2 Input Impedance and Radiation Resistance of Short Antenna Doubly Loaded by Reactances XL of Various Q. v0 = 1 volt, f = 200 MHz, a/AO = 0.00212, d/h = 0.7, h/A0 = 0.075, XL = 1080 Q Radiation Zin = Rin + j Xin Resistance Rin in 100 10 43993 28.51376 -8.65307 200 10.78496 19.82499 -8.28564 300 10.89967 16.92681 -8.19977 400 10 95697 15.47745 -8.16371 500 10.99132 14.60777 -8.l4427 600 11.01422 14.02795 -8.13223 700 11.03057 13.61379 -8.12408 800 11.04283 13.30315 -8.11821 900 11.05237 13.06155 -8.11379 1000 11.05999 12.86826 -8.11035 1100 11 06623 12.71012 -8.10759 1200 11.07143 12.57833 -8.10534 1300 11.07583 12.46681 -8.10346 1400 11.07960 12.37123 -8.10187 1500 11.08287 12.28839 -8.10051 1600 11.08573 12.2159 -8.09933 1700 11.08825 12.15195 -8.0983 1800 11.09049 12.10951 -8.09739 1900 11.09250 12.04423 -8.09658 2000 11.09430 11.99845 -8.09586 w 11 12860 11.1286 -8.08302 33 34 Table 3.3 Input Impedance and Radiation Resistance of Short Antenna Doubly Loaded by Reactances XL of Various Q. v = 1 volt, f = 200 MHz, a/1O = 0.00212, d/h = 0.7, h/IO . 0.1, xL = 850 0 Q Radiation Zin = Rin + j xin Resistance Rin xin 100 18.81215 31.92312 5.67854 200 19.00825 25.566 .01933 300 19.07317 23.44536 .10916 400 19.10554 22.38480 .14916 500 19.12493 21.74840 .17274 600 19.13785 21.32410 .1817 700 19.14707 21.02101 .19727 800 19.15398 20.79369 .02473 900 19.15936 20.61688 .21044 1000 19.16366 20.47574 .21495 1100 19.16717 20.35970 .2186 1200 19.17010 20.26326 .22161 1300 19.17258 20.18165 .22414 1400 19.17471 20.1117 .2263 1500 19.17655 20.05107 .22816 1600 19.17816 19.99803 .22977 1700 19.17958 19.95122 .23119 1800 19.18084 19.90962 .23245 1900 19.18197 19.87239 .22358 2000 19.18299 19.83889 .23458 w 19.20230 19.2023 .25314 . 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Co a x .3386 - 3 ex .N.0 xum1/1 apnaitdmu auazzna SAIJEIBJ 41 Table 3.4 Input Impedance of Short Antenna with Increased Input Resistance and Inductive Input Reactance. h = 0.075A0, a = 0.002121 0 h = 0.1A0, a = 0.002121 0 loading 2, (for d = 0.5h) Loading X Z (for d = 0.5h) Reactance 1n Reactance L in 0 4.3810 - j 608.58 0 8.116 - j 468.287 700 9 641 + j 72.322 600 19.36 + j 190.30 800 11.772 + j 291.736 700 25.43 + j 463.33 900 15.044 + j 592.851 800 36.397 + j 883.67 1000 20.53 + j 1031.74 900 60.135 + j 1614.56 1100 31.028 + j 1730.99 1000 132.122 + j 3201.50 XL Zin(for d = 0.7h) XL Zin(for d = 0.7h) 0 4.381 - j 608.58 0 8.116 - j 468.287 1100 11.59 + j 25.307 900 21.63 + j 89.38 1200 14.77 + j 236.85 1000 29.32 + j 325.86 1300 20.53 + j 569.52 1100 45.945 + j 747.74 1400 33.32 + j 1169.06 1200 98.03 + j 1713.40 XL Zin(for d = 0.9h) XL Zin(for d - 0.9h) 0 4.381 - j 608.58 0 8.116 - j 468.287 .3100 17.943 + j 206.2 2500 23.24 + j 20.865 .3200 27.95 + j 605.7 2600 31.177 + j 210.15 .3300 64.23 + j 1642.44 2700 50.567 + j 588.9 2800 136.93 + j 1725.83 42 short antenna than by the antenna of Sec. 3.3. For the practical non- ideal case, this is not always true, since the powers dissipated in the loading inductors and tuning capacitor are larger than in Sec. 3.3, and therefore, the radiated power may sometimes be reduced more than the former case. 3.5. Conventional Top-Loaded (or End-Loaded) Antenna: In Sec. 3.3, it is indicated that the current distribution along a short antenna with optimum double reactance loading is al- most constant between the loading points, and it is further in- dicated in Sec. 3.2 that the power radiated by the antenna is approximately proportional to the square of the area beneath the antenna current distribution. Therefore, if the loading position is shifted to the end of the antenna, or if the antenna is end- loaded by an optimum reactance, the current will remain nearly con- stant along the entire length of the antenna, and the radiated power will be increased by four times relative to that of the unloaded antenna. Since the end loaded antenna represents an extremely difficult theoretical problem, only an experimental study will be conducted in Chapter 6. Various types of end and impedance loadings will be studied experimentally to investigate the char- acteristics of end-loaded antennas. 3.6 Comparison of Doubly Loaded Antenna with Unloaded_§ase-Tuned Antenna for Radiated Power and Efficiency: In Sec. 3.1, it is indicated that the input power supplied through a transmission system to a double loaded antenna which is 43 tuned to resonance is always greater than that supplied to an unloaded antenna which is base tuned. If the double loading or base tuning inductors are non-ideal (it is assumed that the capacitor for tuning the doubly loaded antenna to resonance is an ideal capacitor), and consequently dissipate a certain fraction of the input power, the efficiency of the antenna should be considered. The efficiency of an antenna is defined as (Pin - Pdissip.) P. In Efficiency = X 100% Therefore, the effiencies of the optimum doubly loaded antenna (only the doubly loaded antenna with increased input resistance and zero input reactance will be discussed in this section, since the efficiency for case with increased input resistance and inductive input reactance is similar to the former one if the tuning capacitor is assumed to be ) ideal) and the base tuned, unloaded antenna designated as (Eff op and (Eff)no’ respectively, are calculated as 2 I (z=d) ( ) (Eff )O = [1 - 2 3 (:L ;P J x 100% ' p Iz(z=0) in op (E ) =[ in)“ J x 1007 ff. no (Rm)no + Rb ° where (X ) (RL)op = ——L—22 = resistance of optimum double loading Q - . . Inductors [XLJOp w1th given Q. O'X U‘ = resistance of base tuning inductor Xb with given Q. Rb: 44 The efficiencies of antenna with fixed dimensions, for both the optimum doubly loaded and the base tuned cases, which result with different Q values for the inductors are indicated in Fig. 3.11. From these numberical results, it is evident that the antenna efficiencies are very nearly equal for both the doubly loaded and the unloaded (or base-tuned) antennas. The power radiated by the doubly loaded antenna is therefore greater than that of the base tuned antenna, since the input power for the former case is always greater than for the latter if the antennas are driven by matched generators. The same conclusion is indicated by the numerical re- /(P sults presented In Fig. 3.12, where the rat1o (Prad.)op rad.)no is plotted for various value of Q for the optimum reactance (XL)op and the base tuning reactance Xb when the antennas are driven by matched generators. It is demonstrated that (Prad )Op is always greater than (P ) under these conditions. rad. no 3.7 Bandwidth of Short Antenna with Enhanced Radiation: The optimum loading reactance for enhanced radiation and the reactance of a fixed inductance loading are indicated as a function of frequency in Fig. 3.13 for an antenna of fixed dimensions. The fixed loading inductance is chosen such that its reactance XL is equal to the optimum reactance [XLJOP at a frequency of 200 MHz. It is observed that the optimum loading reactance is a decreasing function of frequency. The reactance of the fixed inductor, however, is directly proportional to frequency. As indicated in Fig. 3.13, the optimum reactance [XL]Op is 850 0 at f = 200 MHz for an 45 .o acuuavcH no 933255“ aw manage—4 tonne-onmm was 000.25 3.253330 556300 00 muouauauwumu 3.0 .0?” .3332: no 0 000N 000.“ 08 H 00.11. 0% ._ oomz 000 08 00¢ 00~ o . 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Ch ocACCmV x ocAcuav . ocA curse . oACCmV x oAcuao . oA sates aoAcaae w on :A o no :A u o: cw A «0+ 0 on CA .00 GA A mv+ x 00 CA A momA o 0 u A av A mva o 0 u A AC coca - am > > n.0 0.H n.A ou(’P'Ja) o('pa‘a) 47 .Azucmsncmo.o I Av acuosvcH vmme a mo oucwuowwm wafiumoa any zuw3 omHMQEou newumwvmm vmocmscm now mocmuoumm wagvmoq Baewuao Hwowuouomsfi w xocmsvmum «ccmuc< I am: com 0mm owu oma ‘ 4 ‘ + 1‘ mucus 1050.0 I A wocauosuCA vmxfiw m we wocwuommm wcfivwoq wmmo cofiumflvmm vmocwzcm pow wcdnmoq o>wuowwu Bzewuao b 4b m~.m .wfim 00H EU mg I r— ...WN.O I QN OOn oooH coma ooo~ OOmN x aouaaoeal Suxpaoq 1 48 antenna with a = 0.125 inch, h = 15 cm. As the frequency is varied to 200 MHZ‘: 10 MHz, the actual reactance of the fixed inductance (L = 0.676u Henry) varies to 850 0 1:50 Q, while the optimum re- actance varies to 850 O1: 50 O. This implies that the difference between the actual reactance of a fixed inductor and the optimum reactance for enhanced radiation is‘: 100 Q for a frequency dif- ference of i 10 MHz. For a 5% variation in frequency the reactance of a fixed inductor varies by 11.8% from the optimum reactance. This result appears to imply that the bandwidth associated with a lumped inductor implementation of the optimum loading reactance will be quite narrow. The input impedance of an unloaded antenna and an antenna doubly loaded with a fixed inductance of L = 0.676u Henry are compared in Fig. 3 14. It is noted that the impedance of the doubly loaded antenna varies with frequency much more strongly than does that of the unloaded antenna. This implies again that the bandwidth of the doubly loaded antenna with enhanced radiation is relatively small. Z. in of X. in reactive component 2000 1800 1600 1400 1200 1000 800 600 400 200 -200 -400 -600 -800 -l000 -1200 49 2a - 0.00424l O Fig. 3.14 Frequency (MHz) Theoretical Input Impedances for zero and Reactive Loading (L = 0.6763henry) as Functions of Frequency x1. 1 Zjn xL ' I 4' 60 CHF 1 J MD ,’ _._d30.7h_..l ...__ h-o.1>. I u so I if Non-dissipative loading / ..40 inductor with inductance ll L - 0.676u henry - 130 X. 4» 1n 4. 1“ ..20 1. ’d 4:- / +10 / Jr ’ I / ... a— ”:d ‘— ..— -'—: ti : 3 tr A; 3‘ 0 100 120 140 160 240 resistive component III R. of in CHAPTER 4 SHORT ANTENNAS WITH IMPROVED DIRECTIVITY 4.1 Introduction: It is well known that a conventional short linear antenna has low directivity since its radiation pattern has a very large beam- width. In this research, a double impedance loading technique is applied to appropriately modify the antenna current and thereby improve its directivity. In 1943, La Paz and Miller [2] attempted to determine the maximum directivity theoretically available from a line source antenna by solving for the corresponding optimum antenna current distribution. Later, Bouwkamp and De Bruijin [3] pointed out that arbitrarily high directivities might be achieved from a linear antenna by properly adjusting its current distribution. Similar results for antennas of different geometries were also derived by Riblet [4] in 1948. In 1949, Chu [9] indicated that several problems inherently associated with highly directive antennas are an un- usually high q, a narrow bandwidth, and a low efficiency. Although no methods were suggested for realizing the required optimum current distributions, the research of the above investigators implies that various degrees of improvement in directivity, with associated degradation of the radiated power, may be achieved by careful adjustment of the antenna current distribution. It is the object of this research to investigate the possibility of physically realizing the optimum current distribution to improve the directivity of a short linear antenna. This current distribution 50 51 is implemented by doubly loading the antenna by a pair of lumped impedances. Through the use of an optimum impedance loading it is found that an optimum current distribution is realized when the phase of the current is reversed along the short antenna. The directivity corresponding to such a distribution of current is improved sig- nificantly. It is found that such a current distribution results in an antenna having poor efficiency and small radiated power, and that these characteristics are closely associated with the improved directivity. A theory is developed to predict the optimum loading impedance for improving the directivity of a short antenna and to determine its input impedance, current distribution, efficiency, and radiated power in the following sections. 4.2 Radiation Field of a Short Antenna with Improved Directivity: For calculating the radiation fields of a short antenna with increased directivity, a mathematical approach similar to that in Sec. 3.2 is applied. Eq. (3.4) can be used directly subject to the same assumptions and approximations, i.e., u -JBORO h 1 L _ _2.E__ - v _ _ A9 - 2” R0 sine f0 12(2 )(1 2 8 r1 2 02'200329)dz' (4.1) where Iz(z') is the current distribution of eq. (2.24). The well known E and B-fields in the radiation zone of the antenna r r Ee--JwAe r _ l_ r B¢ - v E9 52 are therefore obtained as r - j Co e-JBORO Ee - x0 -E;- A F(Boh,9) (4.2) r j HO e-JBoRo B¢ - A R— A F(80h,9) o o where h I I A = I 12(2 )dz ..... the area below the current 0 distribution of eq. (2.24) . B 2 F(Boh,9) = Sin9(l - K'Cos 9) (4.3) and ...];hziz I l B - 2 [0 602 12(2 )dz . By investigating eq. (4.3), it is observed that the radiation pattern is a strong function of the ratio B/A. From numerical results for the current distribution 12(2) expressed by equation (2.24), it is found that the antenna current has approximately a variable-amplitude and constant-phase distribution along the antenna. This implies that although the amplitude of the current varies from point to point along the antenna, its phase is almost constant except for the possibility of a rapid 180o phase reverse that may be accounted for by changing the sign of the amplitude. By using this constant- phase variable-amplitude approximation, the current distribution 12(2) of eq. (2.24) may be represented approximately by some real function jG f(z) multiplied by a constant phase factor e o, i.e., je 12(2) = f(z)e ° 53 where f(z) may take both negative and positive values. Therefore, the ratio B/A is approximately equal to a real number, i.e., l h 2 -2- f0 (eoz') Iz(z'>dz' K = § = h ; real number. (4.4) I I (z')dz' o z The radiation pattern of the doubly loaded, short antenna is the graphical representation of . 2 F(Boh,9) = Sin9(l - K Cos 9) (4.5) in polar coordinates. Different radiation patterns are therefore obtained by specifying different values for the real constant K. The radiation patterns for K = 0,1, 2, 2.33, 3, 10, 46 and w are plotted in Fig. 4.1 through Fig. 4.4. It should be recalled that the directivity of an antenna is the ratio of the maximum radiation intensity to the average radiation intensity. In other words, the directivity is the ratio between the radiated power Pmax. when the antenna is assumed to radiate with its maximum power in every direction and the total radiated power Prad of the antenna, i.e., £12I . . . _ max. 8 d0 max. Max. radiation intensity_ D(direct1v1ty) P 1 Average radiation intensity (4'6) rad. -— P 4n rad. where dP 4n —— Pmax d0 max %% = Rifio - gr ... power radiated per unit solid angle and 54 0.5 180° Fig. 4.1 Theoretical Radiation Patterns of Short Antennas with K = O, l, and 2. 90° K aw. nam)3mx 2.33 38° 4.48db 90° 3 36° -387db 90° K: 1.0 ofif Fig. 4.2 Theoretical Radiation Patterns of Short Antennas with K = 2.33 and 3. 180 O LIL x 1m. D(db) emax 10 34° 4.62 db 33°12 46 35° 4.3 db 34°23 56 K - 10 Fig. 4.3 Theoretical Radiation Patterns of Short Antennas with K 3 10 and 46. 57 s.w. D(db) emax 36 4.15db 35°16' 180° Fig. 4.4 Theoretical Radiation Pattern of Short Antennas with K.*'¢. S8 -or 1 —o —o* . . S = 3(8 X H ) is the Poynting's vector in the radiation zone. The directivity of an antenna in db is defined as Ddb = 10 log10(D)db. (4.7) The beamwidth of an antenna radiation pattern is defined as the angle between the two half-power points of its major lobes. From eq. (4.2), the total power radiated by an antenna for each Specified K is obtained as 2 n 1 ~ ~* = 2n — x - ' Prad. R0 0 2(E H ) RO Sine d9 TTQ - 2 n 3 2 2 = —§—9 A f Sin 9(1 - K Cos 9) d9 1 o o 4 2 - c1[105 (3K - 14K + 35)] (4.8) g fiAZ where C = O l 2 A o The average radiation intensity is then %; Prad . By differentiating eq. (4.3) with respect to 9, the angle of maximum radiation, emax , is obtained as :1: for OSKSzl' max. 2 = Cos-1./2K+l for K 2 4 max. 3K This means that for K > 4, the major lobe is in off-broadside direction. The maximum power intensity is therefore determined as S9 dP 2.. «r 2.. l -' -°* —— = . = o— X dfl|max. R0R0 Smax. R0R0 2(E H )max. = 1-59 A2[Sin8 (1 - K C0520 )2] 2 X2 max. max. 0 Cl/Zfl for o S K s 4 = (4.9) E-C—l- K4 3 f K >_ 4 n 27K or Therefore, the directivity for each specified K is obtained as 125 2 1 for o S K s 4 D(directivity) g (3K —14K+35) 19 (K- 1) 3 2 9 (3K2-14K+35)K f°r K 4 The maximum directivity for the case of 9 = 900 (or 0 S K S 4) max. is therefore determined from eq. (4.9) by letting 2% = 0. This leads to K = 2.33 as the optimum value. The case for K 2 4 is not discussed further since its major lobe is in off-broadside direction. The directivity in db is obtained by substituting eq. (4.9) into eq. (4.7) for o S K S 4 and K 2 4 respectively, as 10 loglo'lg2 2 1 ] for o S K S 4 (3K -14K+35) Ddb = 70 (K-1)3 (4.10) 10 iogloflg— 2 for K 2 4 (3K -14K+35)K The directivity and beamwidth of the short antenna, as de- termined by eq. (4.5), (4.9) and (4.10) are tabulated in Fig. 4.1 through Fig. 4.4 for the various values of K. For K = o, the 60 radiation pattern is just that of a conventional short antenna (B.W. = 90°), and the directivity is D = 1.7db (emax. = 90°). For K = 1, the major lobe becomes sharper (B.W. = 550), and the directivity is improved from the original 1.76db to 3.424db (Gmax. = 900). When K = 2, although side-lobes appear, it is found that the beamwidth becomes relatively narrow (B.W. = 420) and the directivity is increased to D = 4.4db (emax = 900). For K 2.33, the beamwidth is equal to 380 and the antenna has the highest directivity D 4.48db in the broadside direction (a = 90°). For K 3, the side lobes grow larger and the max. directivity is decreased to 4.387db (emax. = 900). For K = 10, the lobe in the broadside direction becomes a minor lobe, and the minor lobe in the off-broadside direction becomes a major lobe with the maximum radiation in emax. = 33°12’, and with a directivity equal to 4.64db (B.W. = 34°). For K = 46, the broadside field is decreased even more than the case of K = 10, and with D = 4.3db, emax. = 34°23' and B.w. = 35°. As K a m, the broadside field vanishes completely and the radiation pattern has four symmetrical lobes (shown as in Fig. 4 4), each having B.W. = 360, and the directivity is equal to 4.1Sdb (Smax. = 35°16'). Since the most interesting radiation field is the broadside field (emax. = 90°), therefore, it is found that the most desirable radiation pattern is achieved for the optimum value of K = 2.33. But for mathematical simplicity, the value of K = 2 will be used in the numerical examples in the following sections. It is now evident that to have an improved directivity the value of K should closely approximate 2.33. From eq. (4.4), this 61 -h condition requires Jo Iz(z')dz' to be very small, thus implying phase-reversal of the current along the antenna. 4.3 Optimum Loading Impedance for Improved Directivity: From the results of the previous section, it is evident that by properly choosing the real constant K, or by properly adjusting the current distribution Iz(z), given by eq. (2.24), the directivity of a short linear antenna can be significantly improved relative to that of a conventional short antenna. Since the current distribution 12(2) of eq. (2 24) is a function of the antenna dimensions, the excitation frequency, and the impedance and position of the loading, the optimum loading impedance for improved directivity can be de- termined from eq. (4 4) and eq. (2 24) if K is specified and the antenna dimensions and its excitation frequency are given. An expression for the optimum loading impedance [ZLJOp is obtained as follows: Eq. (4.4) can be rewritten as 1 2 h .2 , , h , , 3 80 f0 2 12(2 )dz = K‘fo 12(2 )dz (4.11) It is recalled that the current distribution 12(2) on the doubly loaded short antenna was given in eq. (2.24) as Sin 80(h - ‘21) 12(2) ' FC5(CosBoz - Cos Boh) + FC4 - ’ - h ’ -d j FC -ZSin Boh Cos Bod Cos Boz Cos Bo (Sin Bolz I 2 + Sin Bolz+d|)]. By substituting this expression into eq. (4.11) for a given value of K and carrying out the integration, a simple equation is obtained, 62 after some rearrangement, as FC5D6 + FC4D7 - J FC2D8 = 0 (4.12) where l 2 2 2 2 = 0— + - . - 0 D6 Boh(2 3 80h )Cos Boh (80h 2)Sin Boh 2K(Sin Boh - B hCos B h) 0 O i -22 - D7 — Boh + 2(CCs Sch 1) 2K(l - Cos Boh) _ 2 2 2 2 D8 - 2(Boh - 2)Cos Bod + 2(2 - Bod )Cos 80h - 4K SinZB h Cos a d + 4K(l - Cos s h Cos e d)Cos e h. 0 O O 0 0 Since FCZ’ PCA and FC5 are all functions of the loading impedance Z eq. (4 12) can be rearranged, after a great deal of algebraic L, manipulation, into a quadratic equation for ZL’ the solution to which gives the optimum loading impedance [zLjop’ i.e., 2 _ 1312L + 322L + B3 - o (4 13) where g - + - B1 D6D9(D12D9 D13 + D14D10D9 D14D11 D18D15D16D12) + D18D6D15(D16D13 ’ D17D10D9 ' D17D11) + FC4D9D7(DioD9 + ”11’ - 1 D (D + D D D + D D ) 15D8 16D12D9 ' D16D13 17 10 9 17 11 B = D6(2D1 D - D + 2D D D + D D - D D D D 2 2 9 13 14 10 9 14 11 18 15 16 12 ' D18D15D17D10) + FC4D7(2D10D + D 9 11) ' 3 D (D16D12 + D 15D8 17D10) B3 ‘ D6D12 + D6D14D10 + FC4D7D10 63 T 'di D =-———l—————— Sec 6 h Sin 8 (h-d)(Cos B d - Cos B h) 9 30 Tchidr o o o o +-—-‘l-- Sec 8 h Cos 8 d SinZB (h-d) 15 Tidr o o o ca D10 1 - T (Sec Boh - 1) cd T. T (SecB h-l) : idi ca 0 . D11 (—-fif-——- + j Tia) 30 T T Sec Boh Sin Bo(h d)(Cos Bod Cos Boh) cd idr cd (Sec B h-l) T g 0 ca D12 T T (r Tsdi D5 + j DSTsa) cd sdr cd D = (Sec Bob-1) Tca Tidi Tsdi D3 + j D Tca idi 13 3O Tcd Tsdr T 2 T. 4 Tcd idr cd 1dr Tia sdi +1 (————D +1D)] T. T 3 4 1dr cd D = Tsdi D5 14 Tcd Tsdr D ' Sec 80h Sin 60(h-d) 15 30 T. 1dr = d - h D16 C08 80 Cos 80 D D17 = 3 T Sln Bo(h-d) + D16D14 sdr idi D - ———— 18 Ted The quantities T. ., T , T ldl idr T , T , T , T. and FC cd’ sdr sdi ca 18 4 are all defined in Chapter 2. By solving eq. (4.13), the optimum loading impedance is obtained as rv 64 2 = 432/131) :JCBZ/Bl) 4033/31) op 2 [2L] (4. 14) By investigating the numerical results obtained from equation (4.14), it is found that only one value of optimum loading impedance [ZLjop 15 of interest, i.e., 2 -(B /B) -\/(B /B) -4(B ls) The other root to equation (4.14) is extraneous since it does not result in a phase reversal in the antenna current and therefore can- not result in any impogvement in its directivity. The optimum loading impedances for different antenna lengths and various loading positions d are plotted in Fig. 4.5 and 4.6 for the case of K = 2.0. Since the resistive component of the optimum loading impedance is very small (always smaller than 1/10000 of the reactive component), only the reactive component is shown in these Figures, and the optimum loading impedance can be treated approximately as a pure reactance. From these Figures, it is observed that the optimum impedance has its smallest value when the loading position is approximately d = 0.7h, and increases rapidly as the loading position is moved toward either the driving point or the extremity of the antenna, 4.4 Typical Current Distribution on Antenna with Optimum Loading: When a short antenna is loaded by an optimum non-dissipative impedance (inductor), as determined from eq. (4.12) for K = 2.0, at fixed position along its surface, the antenna current Iz(z) of 65 o o A 4H.o z AmNH.o u so zuwhfiuuonwo vw>ouaEH pow moamuuuom weavuog Eaawuao HuuHuouooSH m.¢ .wfim T :\v acouamoa wcaouog o.H m.o m.o “.0 0.0 m.o «.0 n.o N.o H.o o Illwl 0 . . . . . . . 1 o ooo~ T7 11 M11 smzoom u C ox . ooou x mno.ous . ooon . ooom o x nNo.ou£ : ooow A ooow no A d°[1x] aauanosau Surpeoq unwindo 66 o o A KN.0 1 KmNH.0 u 50 muw>fiuumpflo co>oumEH how mucouommm wawoaoq Esawumo Hdufiuuuomsa 0.0 .wom 1 n\v cofiufimoa wcwvuoq D.H ano m.0 5.0 0.0 m.o «.0 m.o Nuo H.0 b n b . ~ p OA ~A~oo.o n 4 Al. .lln. 0 ~ on: oo~ u u r ax nu oA nAA.ons 0‘ ~.0fl: 0x mA.ouL ox mNH.0u£ JV 2: 000a oooN 000m 0000 do [1x] aouunaeau Suipuoq mnmrado 67 eq. (2.24) has the general form indicated in Fig. 4.7. I z 1.91 z=o r~_, z I "1 o I I z=-h z=-d r_._.__._..L90.......1 z=d z=h n o . ._ 0 u I O . _______ _I 1,-90 l.-_____.— lphase Fig. 4.7. Current Distribution on an Antenna with Optimum Loading for Improved Directivity. The antenna current has zero amplitudes at points (z =‘: 20) be- tween the driving point (z = o) and the loading points (z = i d), at which points the phase of the current is reversed by 1800. The total area under the current distribution along the antenna is there- fore almost equal to zero, i.e., A e 0, resulting in a very small input resistance and significantly reduced radiation power. The current distribution of eq. (2.24) for antennas of various lengths with optimum loading impedances [zLjop at several different positions, are plotted in Fig. 4.8, 4.9 and 4.10. 4.5. Input Impedance of Short Antenna with Increased Directivity: By using eq. (2.26), the input impedances of short antennas having both optimum and zero loadings are evaluated numerically as indicated in Fig. 4.11 and 4.12. 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Radiated Power and Efficiency of Antenna with Optimum Loading: By applying equation (4.8) directly, the power radiated by the antenna with optimum loading is obtained as 2 C IAI ; 0 TT 2 2 . 3 Prad. - *12 fou - K Cos 6) Sin 6 de (4.16) 0 Therefore, the radiated power is approximately proportional to IAIZ, i.e., 2 h , , 2 Prad. ~ IAI |f0 12(2 )dz I... area below 12(2). Since the current distribution associated with high directivity re- quires a phase reversal along the antenna, then consequently the integral A = I: Iz(z')dz' nearly vanishes and the power radiated by the short antenna with optimum loading is very small. The efficiency of the antenna with optimum loading is also relatively poor. This can be observed as follows. The efficiency of the doubly loaded antenna is determined in Sec. 3.6 as 2 I (z-d) ( ) Efficiency = (1 - 2 -§* R: 02) X 100% I (z=o) in (XL) 0 ____2£ where (RL)op Q . Since (XL)op has values between 2000 and 7000 ohms for antenna with half-lengths between h = 0.02510 and h = 0.2).o and a = 0.0021210, and Rin has values between 0 and 0.5 ohms while Iz(z=d) and Iz(z=0) are comparable, then unless Q is very high most of the input 74 power to the antenna will be dissipated in the loading inductors. The efficiency is therefore relatively poor. CHAPTER 5 DOUBLY LOADED COUPLED SHORT ANTENNAS The study of coupled, doubly loaded, short antennas involves an investigation of a closely spaced parasitic array of short cylin- drical dipoles. It is the objective of this research to enhance the radiation or improve the directivity and radiation pattern of the short antenna array. Two identical parallel antennas doubly loaded by lumped impedances are investigated in this chapter. The theory developed in this chapter is based on the modified method of King and Wu [6 ], and the investigation of King and Sandler [10]. 5.1 Geometry of the Doubly Loaded Short Array: The geometry of two identical, doubly loaded, short antennas is as indicated in Fig. 5.1. Antenna 1 and antenna 2 are assumed to be constructed of two perfect conductors of the same radius a and length 2h (h is the half-length of antenna), and the distance be- tween these two antennas is b. Two ideal, harmonic voltage sources of equal angular frequency w and potentials V10 and V20 excite the cylinder 1 and 2, respectively, at their centers 2 = 0 (the antennas are assumed to be oriented parallel to the z-axis). In antenna 1, the two identical lumped impedances ZLl are loaded symmetrically on the antenna surface at z = d and z = -d. A second pair of identical lumped impedances ZLZ are loaded on the surface of antenna 2 as indicated in the figure. The gaps in the cylinders at the location of the sources and the loading impedances 75 I12(2) 10 2a - 12 L1 76 T T \J —z--h 22 26 26 _L .1. ——Z a d— T T Z 26 26 _L .1. 7:0 T T 26 26 _.L ___z --d— E 2.. Ffi-Za 122(2) A) Fig. 5.1 The Doubly Loaded Coupled Short Antennas 77 are assumed to be of length 26. Since both the sources and the lumped loading impedances are considered to be idealized point elements, then 6 is assumed to approach zero in the subsequent mathematical analysis. The dimensionsof interest for both antenna 1 and 2 are the same as those for the isolated linear antenna in Chapter 1, i.e. 80h h/AO = E;—'S 0.1 h >> a B a << 1 o b > a & b < h ... closely spaced array where 10 is the free-space wavelength and so is the correspond- ing wave number. As a result of the above thin-wire assumption, and due to the symmetry of the cylinders, the currents on both antennas will flow primarily along the axial or z-direction, i.e. 11(2) = 2 112(2) ... axial antenna current in element 1. 12(2) = 2 122(2) ... axial antenna current in element 2. These dimensional restrictions and axial current approximations lead to a well known result [7 ] that 112(2) and I (2) can be assumed 22 to be concentrated along the axis of cylinders l and 2, respectively, when calculating the vector potential at their surfaces with neg- ligible error. 78 5.2 Boundary Conditions for Calculating the Antenna Current Distributions: The currents 112(2) and I2 (2) on cylinders 1 and 2, 2 respectively, are symmetric about the cylinder centers (2 = o) and must vanish at either of their extremities (z = 1h). A pair of boundary conditions for each antenna current may therefore be ex- pressed as Imz (Z) = Imz (-2) for m = 1,2 . (5.1) Imz(:h) = 0 Since the tangential component of electric field should be continuous at the antenna surfaces as i + a _ Emz(r a ) - Emz(r a ) for m 1,2 (5.2) i + where Emz(r = a ) is the induced electric field just outside the I + O O D . surface of cylinder m at r = a which is maintained by the a - , currents and charges on both antennas, and Emz(r = a ) 18 the applied electric field just inside their surfaces at r = a . 5.3 Integral Equations for Antenna Current Distributions: The arguments and mathematical procedures for obtaining the 2nd-order inhomogeneous differential equations for the vector potential at the surfaces of antennas l and 2 are same as Sec. 2.3. The equation (2.10) can be applied directly here for two vector potentials Alz(z) and Azz(z): they are 79 8 A12 2 jBo 322 + 60AM - T {-V106(z) + 112(d) ZL1[6(z-d) + 6(z+d)]} (5.3) 2 . 2 622 + 60422 - T {-v206(z) + 122(d) ZL2[6(z-d) + 6(s+d)]} (5.4) where A12 is the vector potential at the surface of antenna 1 due to currents on antennas 1 and 2, and A is the vector potential 22 at the surface of antenna 2, due to the currents on both antennas. 1 (id) and 1 (id) are the load currents at the impedance 12 22 elements 2L1 and ZLZ at z = id, and 6(2) is the Dirac delta function. The complementary solutions of eq. (5.3) and (5.4) are obtained easily as = - L ' Alz(z) v (CIICOSBOz + C1281n802) C = .. . ' A22(z) %; (CZZCOSBOZ + CZISinBOz) where V0 is the velocity of light in free-space, and 011, C22, C12 and C21 are arbitrary constants. The particular solution of equations (5.3) and (5.4) are found as v z I (d) p a _ .1. .19. - - 44.1.12!— - Alz(z) v0 [ 2 SinBOIZI 2 (SinBolz d| + Sin80|z+d|)] v z I (d) 432(2) = - 1— Egg SinBoIZI - ———-L2212 (SinBo|z-d| 3‘ Sin50|z+d|>] - v 2 o The general solutions to differential equations (5.3) and (5.4) are thus 80 Alz(z) = A:z(z) +-AEz(z) . V _ ] + . 10 . vCECllCOSBOz C12S1nBOz +'—§- SinBOIzI ZLlllz (d) - -——7f—————(SinBoIz-dI + SinBoIz+dI)] (5-5) 22(2) = 9122(2) + Apz (Z) V20 - 1.. - ___. - V0 [CZZCOSBOZ + CZISIHBOZ + 2 Slneolzl Z I '(d) - ‘Lgagé———(SinBOIz-dl + SinBoIz+dI)] , (5,6) By the symmetry of the antenna currents Imz(z) = Imz(-z) for m = 1,2, and it can be shown that the vector potential is also symmetric about the center of either cylinder, i.e. Amz(z) = AmZ(-z). It is therefore obvious that arbitrary constants C12 and C21 should be equal to zero in order to satisfy this symmetry condition; equations (5.5) and (5.6) therefore become A1z (z) = - i;[cu C0880 z + Véo SinBOIzI - ELLEE—(2(Sineolz--d| + SinBoIz+dI)] (5 7) 22(z) = - i—[CZ 200638 2 +—208inBoIzI-ELZ;3££31(SinBOIz-dl vo + SinBoIz+dI)] (5.8) at z = h, results (5.7) and (5.8) become V Z I (d) 0 _ j_, h +__lg_ , _ L1 12 . h-d Alz(h) vOICIICOSBO 2 SinBOh '——7{—-——{Sin80( ) + SinBo(h+d)]} (5'9) 81 v z I (d) = _ j 20 . L2 22 . A22(h) v0{C22Coseoh +-—§— SlnBOh ..-——3f————{sinso(h-d) + SinBO(h+d)]}. (5.10) By combining equations (5.7) and (5.9), the arbitrary constants is eliminated from eq. (5.7), and an expression for the vector potential difference at the surface of antenna 1 is obtained as . V Z I (d) _ = j__ 10 M L1 12 C Alz(z) Alz(h) vO SecBOhI 2 oz + 2 088oh S02 2 I (d) L1 12 + [ 2 P - U1]Foz} (5.11) where M = SinB (h - I2I) oz 0 g 0 - + a - o - - 0 s02 SlnBoIz dI SinBOIz+dI sinso(h d) SlnBo(h + d) F = 0058 2 - 0088 h oz 0 o P = Sin80(h - d) + SinBO(h + d) = -' A . Ul J vo 12(h) Similary, v z 1 (d) - -L .22 +_1.2_22___ A22(2) A22m v0 seceohI 2 Moz 2 COSBoh $02 2 1 (d) L2 22 . - [ 2 P - U2]Foz} (5.12) where U2 ' -j voA22(h)' According to the dimensional assumptions h >> a and Boa << 1, the Helmoltz integral for the vector potential at the antenna surface can be simplified as the line integral over an axial current distribu- tion with negligible inaccuracy, i.e. 82 u Alz(z) = ZfiIIfih 112(2')K11(z,z')dz' +-I§h 122(2')K12(z,z')dz'] (5.13) 22 4n where u = the permeability of free space and o _ B R e j o 11 K (2,2') = ——— 11 R11 0 I o e-jBOR12 ... Green 8 functions K (2,2') = —- 12 R12 I g I K21(z,z ) K12(z,z ) for two identical linear K22(z,z ) = K11(z,z ) antennas R = R =./ 11 22 (z-z')2+a u A (z) s —9{I§h Ilz(z')K21(z,z')dz' +-I§h 122(2')K22(z,z')dz'] (5.14) 2 ... the self-distance between an observation point on the sur- face of antenna at z and an element of current of same antenna on its axis at z'. R12 = R21 ='/(z-z')2+b 2 ... the mutual distance between an observation point on the surface of one antenna at z and an element of current of another antenna on its axis at 2'. If the left hand side of eq. (5.11) is replaced by the Helmholtz integral expression (5.13), an integral equation in terms of Ilz(z) and 122(2) 13 obtained as H: O h I I I h I I I ZEEI-h I1z(z )Kd11(""’z )dz I I-h 122(2 )Kd12(z’z )dz 1 . V z I (z) z I (d) = 1. SecB h f 10 M + L1 12 _ [ L1 12 v o 2 2 2 O S 0038 h 02 oz 0 P - UIJFOZ} (5.15) 83 where I = I _ I K 11(2,2 ) K11(z,z ) K11(h,z ) 'jBoR11 ’jBoR11h =_e__ _.§._ R11 R11h I = I _ I Kd12(z,z ) K12(2,z ) K12(h,z ) -JBoRIZ -jBoRIZh =5?— -2.— R12 R12h R11h g“ (h-z')2+a2 = 2 2 R12h /1h-z') +b U1 can be also replaced by the Helmholtz integral as U1=-ij (h) o 12 h = -j :n[fhh112(2')K11(h,z')dz' +-f_h 122(2')K12(h,z')dz'] (5.16a) Similary, equation (5 12) becomes EQ{Ih I (z')K (z z')dz' +'Ih I (2')K (2 z')dz'] 4n -h 12 d21 ’ -h 22 d22 ’ v2 2 I (d) z I (d) _j + L2 22 _ L2 22 _ v0 Secso h {% M02 2 SOzCosBOh E 2 P UZJFOZ} (5.17) where K = K and d22 dll’ Kc121 g Kd12’ U2 a -j voA22(h) I I I h I I I = -j :n[fh h 112(2 )K21(h,z )dz +-j_h 122(2 )x 22(11,2 )dz ] (5.16b) 84 Equations (5.15) and (5.17) are a pair of coupled integral equations for the currents 112(2) and 122(2) on antennas 1 and 2, respectively, and are valid for -h S 2 S h. 5 4 Approximate Solution for the Antenna Currents: The application of King and Sandler's method to the solution of integral equations (5.15) and (5.17) is the subject of this section. This method is mainly based on the King's modified method which consists essentially of assuming the current excited on the antenna to be proportional to the vector potential difference (re- ferred to the end of the antenna). By a peaking property of the difference kernels Kd11(z,z') I and Kd22(z)z ) I I _ I Kd11(z,z ) K11(2,z ) K11(h,2 ) N 6(2-2') - 6(h-z') Kd22(z,z') = K22(Z,Z') - K22(h)z') ~ 6(2-2') - 5(h-z') it is found from the left hand sides of equations (5.15) and (5.17) (2) and I (z) for both large and small Bob that the currents I 22 12 may be taken as the form = + . 112(2) Gl Foz + Bl Moz Cl Soz (5 18) = + + 5.19 122(2) CZ Foz 82 M02 C2 Soz ( ) where 61’ CZ, 31’ 32’ C1 and C2 are arbitrary complex constants. 85 Note that in results (5.18) and (5.19) I (z = :h) II O H A N II II 0 1h) such that the boundary condition at the antenna extremities is auto- matically satisfied, and the currents are symmetric as they should be. By substituting equations (5.18) and (5.19) into the integrals at the left hand side of eq. (5.15) and eq. (5.17) and by separating the Green's function into real and imaginary parts, the integrals in the left hand sides of (5.15) and (5.17) can be expressed in a gen- eral form as 2% IEhEGmFoz' + BmMoz' + Cm802.]Kd11(2,z')dz' + :fi’ 1:hEGnFoz' + BnMoz' + CnSoz'JKd12(z’z')dz' = 2% It-1h[GmFoz' + BmMoz' + Cmsoz'JKdllr(z’z')dz. -j ;% EhEGmFoz' + BmMoz' + CmSOz,]Kdlli(z,z')dz' +' gg'f§htcnpcz' + BnMoz' + Cnsoz'JKd12r(z’z')dz' -j Eg-I§h[GnFoz' + BnMoz' + CnSoz'JKd12i(z’z.)dz. (5"20) for m = 1,2 and n = 1,2 but m # n, and where Foz" M , and 02 S , are same as F , M and S except 2 is replaced by 02 02 oz 02 I: I _ h,' K r(2,2 ) K11r(z,z ) K11r( .2 ) = CosBOR1 1 _ COSBoRllh R11 R11h 86 I = I _ I Kd12r(z’z ) K12r(z’z ) K12r(h’z ) = CosBOR12 - CosBOR12h R12 R12h Kdlli(z’z') '[K111(z’z') ' K111(h’z')] = SinBOR11h - SinBOR11 R11h R11 Kd12i(z'z') ‘EK121(Z’Z') ' K121(h’z')] SlnaoRlzh _ R12h R12 SinBoR12 Since (z,z') becomes very large when 2' is near 2, Kdllr it follows that the main contribution to the part of the integral that has Kd11r(z,z') as kernel comes from elements of current I = I I near 2 2. On the other hand, Kd12r(z,z ), Kdlli(z’z ), and (z,z') remain relatively small when 2 = 2'. This su ests 88 Kd121 that the principle contribution to the part of the integral that (z,z'), K (z,z') and K (z,z') as kernel come has d12i Kc1111 d12r from all the elements of current that are some distance from 2. Due to this peaking property of kernel K (z,z') and non- dllr peaking property of kernels (z,z'), K (z,z') and Kc1111 der Kd121(z,z'), the various integrals on the right hand side of equation (5.20) may be verified numerically to have the following approximate representation: 87 h = I I I $-11 Foz' Kd11(z 2 ”)dz Ihh FOZ'EKdllr-(z’z ) + Kdni(z,z )sz Ydu(z)Foz Ydu(o)Foz = YduFoz (5.21) j‘h F0 (2,2 ")dz = 1! (z)F é 1r (0)12 = 1' F (5 22) -h Kd12 de oz de 02 de 02 ° [h MO (2,2 )dz = Y (2)M -h Kdllr dvr 02 (0)11 f B h s E vr 0: or o 2 0 TT der(h - 4 )M02 for 80h > 2 = derMoz (57.23) h d - ( )F ' v F - Y F 5 24 I-hMo Kdlli (2’ z ) z dvi 2 oz dvi(o) oz - dvi oz ( ° ) h . f-ho M2 «d12(z 2 ')dz' Ydf(z)Foz - ‘i’df(o)Foz YdfFoz (5.25) h ° ' d ' = = I: . J.-h Soz'Kdllr(z’z ) z der(z)soz der(o)soz dersoz (5 26) 15h” S Mdll (Z 2 ')d2' ' dei(2)F (o)Fo oz - Y F (5.27) 02 dei dwi 02 h , , . d 3 B I: . I-h Soz,Kd12(z,z ) z ng(z)Foz Yd8(o)Foz deFoz (5 28) where Ydu(z), Yde(2), der(z), dei(z), Ydf(z), der(z), and ng(z) are very nearly constant parameters which may be evaluated as Ydu(z) = (CosBoz - CosBoh)-1{Ca(z,h) - Ca(h,h) - CosBohEEa(Z,h) - Ea(h,h)]} 88 Yde(z) = (CosBOz - CosBoh)-1{[Cb(z,h) - Cb(h,h)] - cossoh[2b(z,h) - Eb(h,h>]} wdvr(z) = [Sineo(h-|z|)]'lgb{31nsoh[ca(z,h) - Ca(h,h)] - Cossoh[sa(z,h) - Sa(h,h)]} vdvi(z) = (CosBoz - Coseoh)1§gn{31neoh[ca(z,h) - Ca(h,h)] - CosBoh[Sa(z,h) - Sa(h,h)]} Ydf(z) = (CosBOz - Cossoh)'1{51nsoh[cb(z,h) - Cb(h,h)] - CosBoh[Sb(z,h) - Sb(h,h)]} vdwr(z) = [SinBOIz-dl + SinBolz+d| - SinBo(h-d) - Sinso(h+d)]‘1 x &h{Da(z,h) - Da(h,h) - [SinBO(h-d) + SinBo(h+d)][Ea(z,h) - Ea(h,h)] + Fa(z,h) - Fa(h,h)} wdwi(z) = (CosBOz - Cossoh)1$zn{na(z,h) - Da(h,h) - [SinBo(h-d) + sineo(h+d)][Ea(z,h) - Ea(h,h)] + Fa(z,h) - Fa(h,h)} wdg(z) - (CosBoz - Cossoh)’1{nb(z,h) - Db(h,h) - [SinBo(h-d) + SinBO(h+d)][Eb(z,h) - Eb(h,h)] + Fb(z,h) - Fb(h,h)} and where h e-jBoRll Ca(z,h) = f—h CosBoz' E—_ dz' 11 h e-jBoR12 Cb(z,h) = f-h CosBoz' Ef— dz' 12 h e-jBokll Sa(z,h) = I-h SinBolz'l E_— dz' 11 h e-JBORIZ = - I __ I Sb(z,h) f_h SlnBo|z | R dz 12 -JBoRll h e E (z,h) = I ‘-- dz' a -h R11 -jaoR12 h e E (z,h) =j‘ —— dz' b -h R12 h e-JBoRll B ' ' _ ! Da(z,h) f—h SinBOIZ -d|R dz 11 h e-jBoRIZ = - I_ , I Db(z,h) j_h SlnBo|z le dz 12 h e-jBoRll = ' ' _ I Fa(z,h) f-h Sinfiolz +d|R dz 11 h e-jBoRIZ z - I __ I Fb(z,h) I-h 31n50|2 +d|R dz , 12 For Ca(h,h), Sa(h.h), Ea(h,h), Da(h,h) and Fa(h,h), the R11 is replaced by Rllh’ and for Cb(h,h), Sb(h,h), Eb(h,h), Db(h,h) and Fb(h,h), the R is replaced by R Equations (5.23), (5.26) 12 12h. and the real part of equation (5.21) are based on the characteristics l . of kernel Kd11r(z,z ), i.e. h I I K d ~ J‘-h Foz' d11r(z’z ) z Foz h I d ' ~ M I-h Moz'Kdllra’z ) 2 oz h ' I I-h Soz'KdllrW’z )dz ~ 802 and equations (5.24), (5 25), (5,27), (5.28) and the imaginary part of equation (5.21) are based on numerical considerations. It is found numerically that these equations are approximately proportional to the shifted cosine function Foz' The essentially constant 9O Parameters Ydu(2) a Yde(z) I der(2) 9 YdVi(Z) 2 der(2) I Ydf(z) : ydWi (Z) and ng(z) can be replaced approximately by their values at z = o, X . g E. e - —9 while der(z) der(o) for 80h S 2 and der(z) der(h 4 ) W > - . for Bob 2 Substituting equations (5.21) through (5.28) into equation (5.20), the vector potential difference is therefore obtained as 4n _ u—{Amz(z) - Amz(h)] - ((2vadu + Gn‘f +j B ‘t’ + B‘Y +j c “1,de de m dvi n df i + + + Cn‘ydgfl‘oz YduerMoz dercmsoz for m = 1,2, n = 1,2, m # n. (5.29) Equations (5.16a) and (5.16b) also can be expressed as I I I h I I ;-Am (h) J] h1mz(z )K11(h,z )dz +-f_h Inz(z)K12(h,z )dz 0 = G Y + B Y + C Y + A Y + B Y + C Y m u m v m w n e n f n g for m = 1,2, n = 1,2, m f n. (5.30) where Y = C (h,h) - E (h,h)CosB h u a a o *6 II c (h,h)SinB h - s (h,h)CosB h a 0 a 0 >6 ll Da(h,h) - Ea(h,h)SinBo(h-d) + Fa(h,h) - Ea(h,h)SinBO(h+d) we I Cb(h,h) - Eb(h,h)CosBoh 06 I Cb(h,h)SinBOh - Sb(h,h)CosBoh ~€ ll Db(h,h) - Eb(h,h)SinBO(h-d) + Fb(h,h) - Eb(h,h)SinBO(h+d). 91 Instead of substituting equation (5.29) into equations (5.15) and (5.17) as in Chapter 2, it is expedient to substitute (5.29) directly into differential equations (5.3) and (5.4). This procedure results in the following general result: L1'o d2 2 ...... — + - 4” (dz, so )[Amzm Ammo] d2 2 = — + - + ° (dzz Bo ){(Gmwdu + GnYde J Bm‘ydvi + BnYdf + ' Y + + ‘i’ J Cm dwi Cnng)Foz dVerMoz + dercmsoz} -janBO =-—————— - - ‘ + £0 [Vm06(z) ZLmImz(d)[6(z d) + 5(z.d)]} A - -E B 2A (h) ... for m = 1,2, n = 1,2 but m # n. “'0 O mz (5.31) By differentiating the F , M , and S , the various delta function oz 02 oz terms are obtained as 2 -d—2 F = ’8 C088 2 CZ O 0 dz d2 2 ——E M = -28 C038 h 6(2) - B M OZ 0 O 0 OZ dz d2 2 3:3 302 e -30 (31n80|z-d| + SlnBo|z+d|) + 280[5(z-d) + 6(z+d)lo Substituting the above results into eq. (5.31) and equating the co- efficients of the corresponding delta function terms, three inde- pendent equations are obtained as 92 (B Y CosB h = L—23 v (5.32) m dvr 0 Co mo w = j 3T3- 2 1 (d) 5 33 m dwr Co Lm mz ( ° ) r . COSBth‘ydqu + YdeGn + J deiBm + Yden + deicm + ngcnJ + ‘i’ c Esme (h+d) + SinB (h-d)] = 5‘51 A (h) (s 34) L dwr m o 0 no mz ‘° where m = 1,2 and n = 1,2 but m ¢ n. From eq. (5 32), the arbitrary constants B and B are completely l 2 determined by setting m = 1 and 2, i.e. iZn v10 B = = V D (5.35) 1 Co derCOSBOh 10 o 82 = V20 0 (5°36) where D = J 2T7 o QOderCosBoh Constants C1 and C2 can be expressed in terms of 112(d) and 122(d), respectively, by setting m = 1 and 2, i.e. C =12:le I (d) (5.37) 1 Q Y 12 o dwr Z 217 L2 . C2 j C Y 122(d) . (5.38) o dwr Since 112(d) = G1(CosBod - CosBoh) + BISinBO(h-d) + C (SinZB d - 28in8 hCosB d) (5.39) l o o o 93 122(d) = 62(CosBOd - CosBoh) + BZSinBO(h-d) + C2(Sin28 d - 28inB hCosB d) (5,40) 0 O 0 then C1 and C2 can be expressed in terms of G1 and GZ’ respectively, i.e. = + C1 G1D1 D2V10 (5’41) C2 = 6203 + D4V20 (5.42) where jsz D1 = C T (CosBod - CosBoh) 0 c1 jznz D = ————L1 D SinB (h-d) 2 Q T o o 0 cl = - ' - ° 1.2—1: Tcl der (SinZBOd ZSinBOhCosBOd) Co 2L1 j2fizL2 D = (CosB d - CosB h) 3 C T o o 0 c2 jznz D = L2 D SinB (h-d) 4 C T o o 0 c2 T = Y - (SinZB d - 23in8 hCosB (1)12E Z . c2 dwr ‘ o o 0 CO L2 Equation (5.34) can produce two independent equations for the cases of m = l and n = 2, and m = 2 and n = 1; they are - + + ' + + COSBohEYduGl YdeGZ J deiBl Ydez + j deicl ngczJ . . _ 33 . + derCl[SinBo(h+d) + SinBO(h-d)] - ”o Alz(h) (5 43) 94 CosBohE‘i’duGz + Ydecl + J deiBZ + Ydel + j deicz + ngCI] . . 4n + wdwrczbmeomw) + SinBo(h-d)] ”o A2201). (5.44) Substituting equations (5.30), (5.35), (5.36), (5.41) and (5.42) into (5.43) and (5.44), two equations of G1 and CZ are obtained as GlTsl + GZT$2 = WIVIO + w2v20 (5.45) GIT$3 + GZT$4 = W3V10 +W4V20 (5.46) where TSl = CosBOhCYdu + j deiDI) + 2D1derSinBOhCosBod - (Yu + Dle) T$2 = CosBOhCi’de + ngD3) - (Ye + DBYg) TS3 ‘ CosBOhCi’de + ngDl) - (Ye + DlYg) T54 = CosBOhCYdu + j deiD3) + 2D3derSinBOhSinBOd - (Wu + D3Yw) W1 = DoYv + DZYw - j(DO‘i’dvi + deiD2)COSBOh - ZderDZSinBohCosBod wz = DoYf + D4Yg - (Do‘i’df + ngD4)CosBOh w3 = Dowf + DZYg - (Do‘l’df + ngD2)COSBOh W“ = DOYV + D4Yw - j(DOYdvi + deiD4)COSBOh - ZderDASinBOhCosBod . From equations (5.45) and (5 46), G1 and G2 are determined as c =wv, +wv (5.47) (5.48) C) II S < + 2 < 95 where = w3T52 - w1Ts4 5 Tssz3 ' Ts4T81 w = w4TSZ - w2Ts4 6 TsZTSB - Ts4Tsl w z w1Ts3 ' w3Tsi 7 - ’ T$2Ts3 Ts4Tsl W = w2Ts3 - w4Tsl 8 TsZIs3 - Ts4Tsl Constants C1 and C2 are therefore also determined as = + C1 (“501 1’2”10 + W6D1V20 C 2 ' w7D3V10 + (WBDB + D4>V20 Finally, the approximate solutions for 112(2) and 122(2) are completely determined as + + I12(2) GlFoz BlMoz Clsoz [wrF + D M + (WSD + + 5 oz 0 oz D2)So (Foz + DlSoz)w6v20 (5.51) l zJVIO : -+ 122(2) G2Foz BZMOZ + CZSoz + (Foz D3Soz)w7VlO +WF+M+ D+ . E 8 02 Do 02 (NS 3 D4>Sozlv20 (5.52) Equations (5.51) and (5 52) express the.antenna current distributions in terms of the antenna dimensions, their excitation frequency, and the impedance and position of the double loadings. 96 5.5 Input Impedance of the Antenna Coupled with a Doubly Loaded Parasitic Element: The input impedance of antenna 1 is defined as V Z. = ———lQ:—— = . + j X. . in 112(2-0) in in If antenna 2 is a doubly loaded parasitic element with zero driving potential or V20 = o, the input impedance of driven element 1 is obtained directly from eq. (5.51) as _ -1 (Zin)v20=o — [wSFOZ(o) + DOMOZ(0) + (wSD1 + D2)Soz(o)] . (5.53) When antenna 2 is doubly loaded parasitic element center loaded by an impedance 20’ V20 = 422(0) 20 and the impedance zin of the driven element is obtained from eq. (5.51) and eq. (5.52) as (2 ) - {1 + [w8FOz(o) + DoMoz(o) + (W8D3+D4)Soz(o)]zo} in I. v20 I20(°)7‘o x £w5F02 + DoMoz(o) + sozT1 {1 + [w8FOz + DOMOz]zO + [Foz(o) + n soz(o>][FOz(o> + DBSoz(o)]zO}'1. 1 (5.54) 5.6 Radiation From Coupled Short Antennas: The well known radiation fields in the far zone of a linear antenna system are given by r r E = - A a J m e r 1 r = -—-E B¢ v 6 97 in terms of spherical coordinates. 10.95”) zone due to the antenna currents 112(2) of eq. (5.51) and 122(2) The vector potential A;(z) at point P(R in the far of eq. (5.52) can be expressed as -jR -jR u 1 2 r = - _2 . h . e . h , e d . “9(R10’9’Q’) 4n SineU-h I12“ ) R1 dz +I-h 122(2 ) R2 2 (5.56) where R1 and R2 are distances between an observation point in far zone and the source points on antenna 1 and antenna 2, respectively, and R10 and R20 are the distances between the center points of the antennas and the point P as indicated in Fig. 5.2. ant. 1 ant. 2 Fig. 5.2. Geometry for Calculation of Radiation Field Since Bob << 1, the distance R can be approximately expressed 20 in terms of R10 as R20 = R10 - b CosY ... for phase factor R ; R ... for amplitude terms 20 10 where Cos? = Sine Coso. 98 By using the same procedures of Sec. 3.2 and taking the three leading terms of the power series of the Green's functions in eq. (5.56), the odd terms of the series integrate to zero due to the symmetry of the antenna currents. Equation (3.4) can therefore be applied directly to eq. (5.56) to yield -jB°R1° h 1 2 2 r = _ .2 . .e__ . - _ I 2 I Ae(R10,9,¢) 2n Sln9{é10 f0 112(2 )(1 2 502 Cos e)dz e-jBoRlO jB bCosY 1 2 2 2 + ——- e ° f I (z')(1 - — 8 z' Cos e)dz' . R10 0 22 2 o (5.57) The radiation fields E; and B; are then obtained as - R r j Co e jBO 10 E = -- B F(B h,9,¢) (5.58) 6 l R o o 10 . -' R r J no 8 J80 10 B = l R B F(8 h,9,¢) (5.59) ¢ 0 10 o where jB hCos¢Sin8 F(80h,9,¢) = Sin8{1 - § 00829 +-§ e O (1 - % 00829} (5.60) l h 2 ,2 , , A 2 Jo Boz 112(2 )dz - h I I B IO 112(2 )dz _h .. C Io 122(2 )dz D - l’fh 522'2 'd'. 2 o 0 122(2 ) 2 Since the phases of the currents on both elements are essentially constant, it is possible to write 99 % a real constant K1 [same argument as in eq. (4.4)] can: real constant K2 [same argument as in eq. (4.4)] E; J's B K3e where a is the phase difference between 112(2) and Izz(z), and K is a real number. 3 Equation (5.60), therefore, becomes jBObCos¢Sin0 . 2 - F(Boh,8,®) = Sin6{1 - K1Cos 9 + KBeJa e (1 - K2C0826)}. (5.61) Equation (5.61) will be used in the discussion of Section 5.8. 5.7 Enhancement of Radiated Power: The average power flow at P(R10,9,¢) is obtained as a 2 § —lee(sarxo r"‘)=l|EgI is (5 62) av. 2 9 H¢ 2 Q ' O 0 From equations (5.58) and (5.61) E; is obtained as r . j Co e-JBoRlO 2 ja jBObCos¢Sin9 2 £9 = 1 -§—— B sinefl - K1Cos e + K3e e (1 - KZCos 9)} o 10 jBObCos®Sin8 . and since it is assumed that Bob << 1, then e = 1 and r j go e.JBORlO 2 la 2 E = -——— B Sine{1 - K Cos e + K3e (1 - K Cos 9)}. (5.63) 9 10 R10 1 2 Since Boh << 1, unless the phase of the antenna current is reversed h I I h 2 2 >> ' ' ' f0 112(2 )dz I0 802 112(2 )dz h I I) 112.2 II Io 122(2 )dz >'Io Boz 122(2 )dz 100 this means that K << 1, K << 1 and eq. (5.63) can be approximated 1 2 as r j Co e-JBORIO Ee - 1 E—— (B + C)Sin8. (5.64) o 10 The total time-average power radiated by the antenna is thus given by —+ 2 . |Er| Prad‘ = 2171110 I:% 2 ’f “r Sine d9 0 =54; IBICIZ. (5.65) 310 Equation (5.65) shows that the power radiated by the coupled antenna 2 is roughly proportional to ‘3 + Cl . Since B and C are simply the areas under the current distributions along the antennas 1 and 2, respectively, the radiated power Prad may be enhanced by maximizing the area under the currents 112(2) and 122(2) while at the same time adjusting them to have a minimum phase difference. This can be accomplished by appropriately choosing the loading impedances XL1 and XL2 located at the fixed positions (11 and (12 along the surfaces of antennas 1 and 2. The current distributions I (z) and 122(2) for antennas 12 which are optimumly loaded with optimum reactances {XLljop and [XLZJOP to have maximum areas and minimum phase difference, are plotted in Fig. 5.3 through Fig. 5.6 for various antenna spacings. From these figures, it is observed that the typical forms of the current distributions on the coupled antennas loaded with the optimum reactances exhibit nearly uniform amplitude distributions between 101 .onao.o u av mwcficuog uunuo uo uunu suds vuuamaoo acaufivaoo doHuuwvum vuocnsam cu unaccommouuoo mascoua< umamaoo covaoq >Hn=oa :0 acausnauuman ucouuau Huofiuuhousa n.m .wwu ox\u newuauoa 36 8.16 8.6 86 no.6 echo 36 No.6 8..o o . . o c o I max c o I 46; s8 . c 8: I SJ .HO ‘|l" I I c 8: I 3x u N o c 82 I Saw sow c 63 I 3x . s6 IINIIIIIII ...IIIII. NH ITOIO \ ‘\ A NHH \\\fl4 a \ \ H \ \ \ \ w o \ ill 3 N as .64 0 Ex $386 I 3 Ir . q H 7: Es: Ill . n M. T ofi6 I s ..I _ Tll £6 I 6 Y- oqln 1.? 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LI 0. n 1% 30".: ll HIV um: 00m I u . _I oHH6 I s H . m T £6 I H. 1.. V... 0.1 q 4+7 ENNHIIIII _ 104 . H9866 I 0 $533 3:8 we :5 5H: oousasoo aoEHocou eoHusHosm 382.6 3 mcuocomuouho0 mucaouc< 039.50 026on— hanson HHo H5333: 30 000.550 Huuauouour—H 06 .wum o in aoHuHmon H6 86 86 B6 86 86 86 86 86 86 o . I. I . . I x . 6 NH H N f If new . N6 u HOW I | I I I . m TI .30 M . 66 m ' | ' I I I | l ' I I | ' J I . a u a :06 .m I .l- a I. n P a o I w o WU I I I I o.H H qu 0 ~11 ENHH [lo HZV Na: 68 I u . _ Hx o .3386 I 3. _ w HH6 I s or T £6 I H. IL Iflfl E ENNH II _ 3x 105 the loading points. Under the optimum condition of enhanced radiation, the shapes of I 2(2) and 122(2) are quite similar and the phase 1 difference between them is minimum as shown in Fig. 5.1. Thus, the radiated power under this condition can be enhanced to about four times that of an isolated antenna. The input impedances of the driven antenna coupled to a parasitic element, both loaded with the optimum reactances, are determined from eq.(5.53) and shown in Table 5.1. It is indicated that under the optimum condition of enhanced radiation the input resistance is increased and the input reactance is reduced as com- pared with the case of an isolated loaded antenna. 5.8 Improved Directivity: For simplicity, only the case of g = g' is considered in this section. Under this condition, eq. (5.61) can be expressed as F(Boh,9,¢) -"= Sin9(1 - K Cosze)[1 + K3ejo’(1 + j BobCos¢Sin9)] (5.66) l jB bCos¢Sin9 . since e O = 1 + j BobCos¢Sin6 for Bob << 1. If g = K3eja is not close to -l, the radiation pattern is similar to that of a single doubly loaded antenna for improved directivity as discussed in Chapter 4 and no further discussion is needed. How- ever, if C = -B can be implemented eq. (5.66) becomes 2 , 2 F(Boh,9,¢) = j Bob(1 - K1Cos 9)Cos¢ Sin 9. (5.67) The radiation patterns in the plane of m = o are plotted in Fig. 5.7 for various values of K The directivity and beam- 1. width are also indicated in the same figure. By comparing with the 106 0H.Hm on.~¢ own cg no.0 com coo e~.~ nw.o¢ cos oK mo.o can own h~.aoe- w~.n coca ox mo.o o o an.mao n.5na oeN ax no.0 com ooHH 8.5 8.8 on 0‘ 8.0 o8 8... om.~o¢- nn.o oONH ox mo.o o o 3&3 $6... on? 04 86 88 82 8.: - 8.8 on ox 88 8a o8 Ha.w¢¢- mm.e onaa ox mo.o o o $.33 3.4.x: one... ox 88 8: 8: no.m-- -.- oo ex so.o ones onm ma.nnm- ow.~ owaa oa Ho.o o o 5." 5“ Auwuwwuwwwwnmuo “mowuuusfimuw so as can a + cam - as _~¢ - Ho_ n Ngx flax O O .nn.o u u can K~.o u a . KNaNoo.o - u .umz oo~ - u AquNH vac A~quH coo3uon mucuuwmuuo wuwnm can ucoswam oiuuuuunm vuvuoa nu“: vudasoo accuuc< au>aun mo moocuvanH uDACH .H.n wanna. 107 Fig. 5.7 Theoretical Radiation Patterns of Coupled Short Antennas with K = 0, l, 2 and a under Conditions of K = K2, K = 1 and a - n. (¢ = o) 1 3 108 radiation patterns shown in Chapter 4, it is observed that the doubly loaded coupled array has a higher directivity than that of a corres- ponding single antenna if the condition of Ih I (z')dz' = ~fh I (z')dz' o 12 o 22 is met. 5.9 Discussion: The approximate theory developed in this chapter for the doubly loaded coupled antennas has been checked by the existing theories [ 5] for the case of Z ' Z = 0. When 2 = Z Ll ” L2 L1 L2, the theory is still quite accurate, however, the accuracy of the theory starts to de- crease when the difference between ZL1 and 2L2 increases. The reason for this discrepancy is due to the completely different current distributions on the antennas when le is greatly different from 2L2' CHAPTER 6 EXPERIMENTAL STUDY OF SHORT ANTENNA WITH HIGH DIRECTIVITY OR ENHANCED RADIATION An experimental study of doubly loaded short antennas (both for single and coupled antennas) is presented in this chapter. In order to compare these experimental results with the theoretical results presented in the previous chapters, an antenna of particular dimensions, a = 0.125 inch and h = 15 cm, which was used in numerical calculation is used in the following experiments. The current dis- tributions and input impedances of an antenna with different optimum impedance loadings at proper positions along the antenna are measured and are compared directly with the corresponding theoretical results. The excitation frequency is usually fixed at 200 MHz. In addition to the doubly loaded antennas, experiments have also been conducted to study the cases of an end-loaded antenna and an antenna with double impedance and end loadings. 6.1 Experimental Setup: The experimental setup for measuring the current distributions and input impedances of the antennas is shown schematically in Fig. 6.1. Photographs of the inside and outside views of the anechoic chamber are shown in Fig. 6.2 and Fig. 6.3. An 8' wide, 6' high, 6' long anechoic chamber was constructed with wooden frames enclosed completely with an aluminum ground plane on one wall and B.F. Goodrich type VHF-8 microwave absorbers covering the remaining five walls. A driven linear antenna (monopole) and 109 110 door anechoic chamber R.F. absorber covers 5 walls : 8 x 6 x 6 ft. 4 ‘ impedance loaded driven ‘ monopole and array element 4 4 current probe‘ aluminum ground plane 8 x 6 ft. x 1/8 inch thick T 4 11d “ IL 1 Int 1 k Hz 200 MHz amp. mod. I for tuning probe for S.W.R II 200 MHz + 1 k Hz amp. mod. minature 50 0 coaxial cable I‘d: I coupling ___./’/ collar 500 slotted sec. I L°-L 1 1—1 1* variable phase - delay 10 db directional ’/// variable coupler attenuator miniature 50 Q T ._ coaxial cable ‘—\\N , variable short “" for tuning LE l°§J I Fig. 6.1 Experimental Setup Fig. 6.2 The Inside View of Anechoic Chamber II I. Fig. 6.3 The Outside View of Anechoic Chamber 112 array element are simply the extensions of the movable centerwires of the coaxial lines connected to the ground plane. Thus, the antenna length can be adjusted freely by sliding the centerwires inside of the coaxial lines and into the anechoic chamber. The driven antenna is excited by an R.F. OSC. at 200 MHz and with the square wave amplitude modulation of l KHz. The coaxial line which excites the antenna has a characteristic impedance of 7S 0 and its outer con- ductor has physical dimensions of 1 inch outer diameter and 0.875 inch inner diameter. The outer conductor of the coaxial line is directly connected to the ground plane. The center conductor has a diameter of 0.25 inch and its free end, which protrudes the ground plane, serves as the antenna. The lumped impedance is mounted on the antenna as indicated in Fig. 6.4. Since the antenna is separated by a piece of insulating material at the loading position, the loading impedance is actually the parallel combination of the externally mounted inductor L (non- ideal inductor) and the unknown stray capacitance C existing at the loading location. The loading impedance may be determined from the following simple circuit. The impedance Z at angular exciting L frequency w is obtained as L Z = R + j w L 0 -Z L l+ij(R+ij) L R wL where R = —— . J Q The frequency can be adjusted to make the current minimum at the loading point. At this frequency, the suceptance, 1/ZL becomes 113 ground plane exciting coaxial line inductor probe teflon support teflon Support Fig. 6.4 Structure of Monopole Antenna solder joint to center conductor solid wire .030" solder joint [/r—SOO Microcoax line Fig. 6.5 Structure of Current Probe 114 2 zero. This critical frequency can be determined to be mo 8 (L-RZC)/L20. The stray capacitance C can then be expressed as C = L/(R2 +-w:L2) and ZL is determined as (R + ij) (R2 + (”21.2) z=RL+jX= - L 2 2 2 L R+L(w:-w)+ijR (6.1) The current probe, which is connected to a flexible 50 O coaxial line passing through the hollow center conductor of the excitating coaxial line to the instruments outside the chamber, is supported by a plastic guide in the antenna slot and can be moved freely between the driving point z = o (the point at the ground plane) and the loading point z = d. The detailed construction of the current probe is shown in Fig. 6.5. The relative amplitude of current can be measured by moving the current probe along the slotted antenna. The phase of antenna current is obtained by comparing the probe signal with the reference signal from the R.F. Oscillator. A charge probe is inserted into the region between the outer and inner conductors of the exciting coaxial line. This probe is supported by a movable carriage and can be moved along the slotted outer conductor of the exciting coaxial line. The standing wave ratio and the phase shift of the wave pattern in the coaxial line can be measured by this charge probe. The input impedance of the antenna can then be determined as, 1 - j S taanLmin ° X 2 (6.2) 2 . = 1n S j taanLmin. c where S is the standing wave ratio and Lmin is the distance between the first voltage minimum and the antenna driving point 115 and Z is the characteristic impedance of coaxial line. c 6.2 Doubly Loaded Short Antennas: The experimental investigation of the current distribution and input impedance of an antenna both for enhanced radiation and improved directivity is presented in this section. An antenna, having the dimensions of a = 0.125 inch and h = 15 cm, is loaded with a reactance XL of various values at d = 0.7h and excited at a frequency of 200 MHz. The loading impedance ZL = RL + j XL, determined from eq. (6 l), is plotted in Fig. 6.6 in such a way that XL is expressed as a function of RL for the variance inductors L with the same Q = 75 at f = 200 MHz. The measured value is compared with the theoretical value. From here on, the reactive part of ZL’ XL’ will be used as the equivalent loading reactance in the following sections and RL will be omitted in the expression of the loading impedance ZL. 6.2.1 Enhanced Radiation Case In Sec. 3.3 of Chapter 3, it has been shown that the typical current distribution of an antenna with optimum loading reactance [XL]op at the position of z = d for enhanced radiation is a uniform distribution between the loading points which decreases to zero be- tween the loading points and the extremities of the antenna. Under this condition, the input resistance is increased to two to three times that of an unloaded antenna and input reactance is tuned to zero. The experimental results for the antenna current distributions with various loading reactances of XL = 0, 500, 800 and 900 0, at d = 0 7h, are plotted in Fig. 6.7. By examining these curves and ll6 com 000a (aous30991 Snipso‘).1x OOma OOON OOmN av 117 03000000 coHuwavnm voucmccm cu mcavcommouuoo mCOHuanuumwn ucouuao accouc< kucosauomxm n.o .mHm o A\N acuufimom 0H.0 00.0 00.0 50.0 00.0 no.0 «0.0 no.0 No.0 H0.0 00.0 s F p J a 4 ‘ O r 52 u i.eu; _A50.S n 5.0 u 0 fl . 4.: ex 0 . AQN¢00.0 u mm b< _J_J- .0.0 r0.0 20 0.0. 1/1 apnlltdms auazano aAinsIaa 'xsm 118 comparing with the theoretical results shown in Fig. 3.3, it is found that the experimental result for X = 800 Q is correlated to that of L the theoretical results for X = 850 0. This gives a quite satisfactory L agreement between theory and experiment. As the loading reactance is increased to 900 O, the current distribution becomes similar to the case discussed in Sec. 3.4. The experimental input impedance of the antenna, determined from eq. (6.2), is plotted in Fig. 6.8 as a function of XL, which is calculated from eq. (6.1). This experimental result is compared with the theoretical results of eq. (2.26) for an inductor with Q = 75 and Q = m. It is observed that the experi- mental results compare well with the theoretical results for the case of Q = 75. The higher experimental input resistance is mainly con- tributed by the loading resistance RL present at the loading points (z =‘:d), since it is indicated in Fig. 6.6 that the loading resistance RL determined by eq. (6 1) has a value greater than that of the theoretical one. The higher input resistance is therefore expected. 6,2,2 Improved Directivity Case In Sec. 4.4 of Chapter 4, it has been indicated that the typical current distribution along an antenna Optimumly loaded for improved directivity has a phase reversal between the loading points (z =':d) and the driving point (z = 0). At the point of phase reversal, the antenna current goes to zero. The total area under the current distribution along the antenna is, therefore, almost equal to zero. As a result, the input impedance has a small input resistance and a very large input reactance. By examining the experimental results of the current distributions shown in Fig. 6.9 Rin Input Resistance 119 I II 'Il l J» Q I 75 ' 300 . ‘ Q 75 3000 *1]. : I: I» I ' a. ‘ I ' 'l I 250 ‘P : I . inSOO s 3 ‘ . .I . , I . \ 2004» J 3 ' ' ”2000 I I 'I | -\ ' \ I» l : I . \ i El ' o: o N 150 H I I . \ 71500 G, C I 8 4 I a: ‘ g I s 1004 I $10003 U 0 8. I a o H 50‘ \ ’ 500 *--500 ———- Rm experimental results xin theoretical results \ L I N O O O Fig. 6.8 Theoretical and Experimental Input Impedances of a Short Antenna as Functions of Loading Reactance XL for the Cases of Q I 100 and Q I O. 120 sanSap u; aand nuaaana .aoauaoaoo >uw>uuoouwa vo>ouaEH cu mcwocoanmuuoo mcofiusnwuumwn uauuuso accouc¢ Housmaauoaxu m.o .wfih on: OOH- and- CON- ox\u cofiufiuom oo.o mo.o b d 11 F L 0.0 L1 Noo . «.o .To.o :5 ///// o.~ \k. \o\\\0\ manna lllll I l. ||||||||||| 33:95 O . «H.o - n w “4 55.0 I v .1 n 4 Li ® 0x¢~ we when quzwcmuumm news cmvuOAuvcm wccmua<_am co maoAusnAuumAn unouuao AoucoEAuoaxm .AA.o .wAm ox\~ :oAuAmoa H. mo. mo. 50. 00. mo. #0. no. No. #0. o 10‘ o 0 10 in v w. 1+ n . o t~.o 50 N I A .fi 50 e I A +.¢.o 800'1— If So w I A ito.o 80 CA I A : Eu NA I A ...m.o A. O.A A T1 aid u L L : IIWI%II L ‘A a 4.. ...: A A h 2 o A lbl I; llfill [ KQNQO0.0 I QN Mm: CON I u I/I apnnxtdms Juaaano BAIJBISJ 'X'Bln 124 .A:x I av moNAm ozoAuu> mo when AmoAuocAAzo Laws voooOA-vcm mucouc¢.co so maoAusnAuqua ucouuso Aauaoaaumaxm NA.o .wah EUQIH EUQH‘H oA\N soAuAmom A. mo. mo. 50. we. no. so. no. No. Ac. 0 Eu N I A I.N.o EU+Vu1~ 4. .83.; 0.2.0" e L 71— F A ... ..x o «swooo.o.u um um: oo~ I u I/I apnnxtdms auaasno aAIJBIaJ 125 EU 50 EU 80 EU EU oA NA b ‘ "D. Go. .muouoaoan ozoAuu> mo moqum qusoqu :uAa oopoOAuucm accouc< so so occausnAuumAn uaouuso AnacoeAumaxm mA.o .wAm o x\u soAuAmoa s. e. ‘ Mo. 71‘ f EH o . AA.O u L o Aemqoo.o I a N am: com I u // an I/I apnnrtdms auaazno BAIJIIQJ 126 .A=s\m u no «names; macauu> mo moxAAo: LuAB povaAuvcm accoua< so so acoAusnAuumAn uauuuao AuuaoaAuoaxu ¢A.o .wAm o A\u :oAquoa n . m a w xv u in u w. o l, 60 ON I A So on u A 80 mN I A C «.0 i iro.o Eu mm 0 A .. i m.o o.A Awfi .oz .ozav was; Accuses no names; a A O ”1 AA.o u L 1* nun: A ®:+\\MI Q 5 LI 4 1 exswsoo.o . .N Ne: oo~ u a 1/1 apnnIIdms nuallna aatqstal em 127 .AEo m.nn I AV numuoadAn oaoAuu> mo moxAon nuAs oomeA-ncm accouc< co co mcowusnAuqun acouuso Aquamawuoaxm nA.o .wAm o I «\NraoAuAmoa A. «o. mo. no. mo. no. so. no. No. Ac. 0 :OH\m I Q L.N :0~\H~ I Q i o :©~\MH I Q 0 I \ 1 I fa. ©H\NA I Q xAAmL mo noumEmAn I a . O.A SA .oz 035 Eu m.nm I muA3 AmuAAw; mo cuwcmA I A r 0:6 u e 1 4| A- E. ... LI . a Aqwqoo.o I am Nmz ooN I w .— .4— 1/1 apnqrtdme nuaaano aaxasxaa °xam 128 12 cm. In Figs. 6.15 and 6.16, it is shown that when the helix diameter is approximately the same as that of the antenna, the current distribution resembles that of the unloaded antenna with an equivalent half length of h' = h + hl’ where h is the antenna length and h1 is the height of the helix. As the diameter of the helix is in- creased to four or five times that of the antenna, the current dis- tribution becomes nearly uniform along the antenna. In Chapter 3, it has been indicated that the power radiated by the antenna is approx- imately proportional to the square of the area under the current dis- tribution on the antenna. Consequently, the circular plates and helixes are desirable end loadings for obtaining enhanced radiation. 6.3.2 Input Impedances The measured input impedances of antennas with various types of end loadings are listed in Table 6.1 through Table 6.5. It is indicated that the input resistance is increased and the input re- actance simultaneously decreased as the length of the bar or the diameters of the helix and the circular plate are increased. By examining the current distributions and input impedances, it is concluded that by using the end-loading techniques, the power radiated by the antenna may be enhanced by a factor of one to four compared with that of an unloaded antenna if the dimensions of the end loadings are appropriately chosen. However, an end-loading is not capable of reducing the input reactance to zero and cannot in- crease the input resistance by a factor larger than four. It is clear that the technique of doubly loading as discussed in Chapters 2 to 4 can accomplish more than the end loading technique. 129 Table 6.1 Input Impedances of a Short Antenna End-Loaded with Rectangular Bars of Various Sizes (L(cm) X t" x 1 mm. thick). a I 0.00212).o h I 0.1).9 , f I 200 MHz Length of 2in - Rm + J X1 (0) Rectangular Bar Rin xin 2 cm 9.336 -424.818 4 cm 10.941 -373 6 cm 11.92 -336.5 8 cm 11.86 -315 10 cm 12.328 -278 12 cm 14g306 -247 Table 6.2 Input Impedances of a Short Antenna End-Loaded with Cylindrical Bars of Various Sizes (k" in. diameter). a I 0.002121o h I 0.110 , f I 200 MHz Length of zin ' Rm + 3 xi}; (0) Cylindrical Bar Rin xin 2 cm 10.018 -408.352 4 cm 11.09 -356.542 6 cm 12.914 ~318.338 8 cm 13.42 -286.43 10 cm 14.024 -254.2 12 cm 15.32 -223 130 Table 6.3 Input Impedances of a Short Antenna End-Loaded with Circular Plates (1 mm thick) of various diameters. Diameter of 2in B Rin + j xin Circular Plate (cm) Rin X1“ 2 cm 6.17;, ~403.254, 4 cm 10.11 -301.9 6 cm 12.552 ~221.778 8 cm 14.538 -l48.66 10 cm 13.076 - 88.634 12 cm 14.57 - 36.18 Table 6.4 Input Impedances of a Short Antenna End-Loaded with Helixes of Various Length3(D I 3/4"). Length of 2in . Rin + J xin Helical Wire (cm) Rin xin 25 cm 9.06 -348.4 30 cm 14.24 -3l8.34 35 cm 14.56 -292.2 40 cm 15.66 -246.92 Table 6.5 Input Impedances of a Short Antenna End-Loaded with Helixes of Various Diameters (L = 37.5 cm). Diameter of 2in I Rin+ j x‘n . Helical (inch) Rin xin 5/16 inch 4.756 -356.7 11/16 inch 13.473 .249,4 13/16 inch 14.870 -244.696 1 1/16 inch 22.848 -l35.6l 131 6.4 Short Antenna with Double Impedance and End Loadings: A short antenna which is doubly loaded and also end-loaded by various loadings is discussed in this section. Since it is difficult to measure the current distribution between the loading point z I d and the end point z = h, only the current distribution between 2 = o and z = d is measured in this experiment. The current distribution between d S 2 S h is assumed to take that same form as that on the antennas of the previous section. 6 4.1 Current Distribution The current distributions of doubly loaded antennas having various types of end loadings at z = h and various loading re- actances XL at d = 0-7h are plotted in Fig. 6.16 through Fig. 6.23. These figures indicate that the loaded antenna can have typical current distributions appropriate to improved directivity (see Fig. 4.7) or enhanced radiation (see Figs. 3.1 and 3.9) if the appropriate corresponding loading reactances XL are mounted at z = d along with various end loadings at z = h It is also in- dicated that the current distribution is mainly controlled by the loading reactance XL. 6 4,2 Input Impedances The input impedances of antennas with various loading implementations are listed in each figure for various loading re- actances X . L 6.5 Doubly Loaded Coupled Antennas: The experimental results for the current distributions and in- put impedances of coupled antennas doubly loaded by various reactances 132 .80 w «c you AquuchA%u u hp covoOAnucm occouc< wovoOA mAnson a so soAusnAuumAn ucouuso AuucmaAuoaxm 0A.o .wAm ox\u :oAuAmOQ A. oo. mo. so. go. no. so. no. No. Ho. o 11 11 a 1. w h n 1. a L. o l 2.93 «+3.33 s8 _ .¢~NH A+o.mohfi_ nmoa _ .. _ na.ooc~ .-oam _ ~¢~Ax_ ~.o r; .aaw ._ \qx _ Nam“ I ..e.o .1 mnoH - ax .fi 000 ON.“ I A“ Iv ”.0 i sow . ax - - . - » —o.a O K~.o u s a n1 _- ea.o - a EU Q I 1— El b Aw A A o JCT. :N x KJNQO0.0 I QN N—E CON I w I/I apnnitdms auaaano aaiastaa 133 .60 CA mo pom AmoAuocAAuo d >9 wovuOAuvcu accouc< oocoOA %A930A u so mcoAusnAuumAA uuouuso AnacoaAuoaxm “A.o .wAm o «\u soAuAuoa A. so. no. so. so. no. so. no. No. Ho. o n .1 w n a n L. .i .. o m.mao q+mm.aofi Ana ~.am~A.A.na.mnm Aswan. : ma.maw n.nn.m~fl mama can! ax, N o a e.o asNH x 4 : e.o mans . ex .5 .. m.o mms u ax sea I ax - - > - - 4 . o.H O 4 Aa.o u g .r H *0 sh.o u a .. _ ...... as 8 rl:N AK 0 fl Asmsoo.o u «N um: com I a I/I apnnxtdme auaaano aaiaataz 'Xm 134 .60 NA no pun AooAuocAAao a ma omowOA-vcu accouc< cocooA AAnson a co mcoAusnAuumAn ucouuso Amocoawumdxm wA.o .wAm ox\~ coAuAaoa .H. .8. mo. .8. .8. 13. ..3. mo. we. 3. o 4 l1 .1 1 4 4 I q 1 J. o ,. RAIN”: rung” _I4j $.32 7282 fi 32 + 3.23 7853 _ 3~ GA _ AN|_ .TN o aaNH u ax I. «so 32 I Ax . .66 .4 a8 .. f a To a z: u 5x . - - - H o; O 1 .25 a c v A: D n- 5.0 .. a .r L .5 2 u 5 ® .. I A a o . If C .. x 4330.0 u «N 22 08 .. a I/I apnnltdms nualano OAIQBIBJ am 135 .nouoEmAv Eu w mo wuuAm uuAsuqu m an vovaA-vcu accou=< wovaA hAnson a co mfioAusnAuuqu ucouuso AuucoeAuwmxu mA.o .wAw o A\N coAuAmoa A. ac. no. no. So. no. so. no. No. Ho. o o J. u . . 1 . . u . 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No. S. o . n . . 0 u U . 0 u o 32 u Ax .~.o . u. 3: u ax u ...}. n a 3 3.30 To... ma: . m 2:5 Tan; 32 m CwN AN .0 o 3 3 m . d I Tr m . m.o P How u x I .. / I m can u 5x . . - o4 . 0 j/ «A .o u c w __1 £6 u a a A 1:11 3 ‘1 EE .— A O . * x 338 o u «N 2.: 08 n a 137 .uoumemAc 80 NA wo oumAm pmAsoqu m xn pocmOAnvcm accouc< vvaOA mAnson o co acoAusnAuumAn usouusu AmocoEAuoaxm Am.o .wAm o A\u coAuAmOQ AeNA u 5x (\I 5N.O oomH a ax.|\\, .ts.c A1 na.~0a «-5m.aw.~ News _ lee.o _ mm-eso .-AM.ao r oonA1_ A 5~ _ 5x4. .lw.o oNN . 5x x «mm u 5x.||L\ - - - - - o.A O 1 (“H.Oflr— .1 ;A.o u e No A A» a F G A I O _ 1. as A x Asmaoo.o u mm um: com a a I/I aanIIdms 3ualln3 BAIJOIBJ OXW 138 .uouoEuAv Bo A.~ mo xAAo: a kn voomOAiocm accouc¢ coomoA hAnson a so coAuanAuumAn ucouuso AnacoeAuuaxm N~.o .wAm O A\~ coAuAmoa A. oo. mo. no. we. no. co. no. No. Ao. o . 1r 1. n L. t. t. u x 0 o 0.02 $2.8 on: 1238 792% SN : n. so -Ns.-~ n3 5 :~.o N AK1 3.: u ax : .Aq.o t 3... mcmA I Ax f One I Ax .Am.o of u 5x - - - o; O .1 £6 u L r _1 L50 u L. 1 [LI _ s 3. . a p: 6 LI 5 O . x Aquoo.o sz OON n A I/I apnqitdms nuaaino 3A139191 'X‘w .uouoamAc Eu m.N mo 3AAom m xn vovuOA ocm accmuc< covoOA %An=on a co mcoAusnAuumAn uaouuso AquamaAumdxm mm.o .wAm O 139 A\~ aoAUAmoa A. mo. mo. so. co. no. so. no. No. Ac. o o NINA a 5x 050A I AAA .A q.naoA fi+onHaAmA Ana 1; .. o.o o~.omm A-sm cmH_ oaoA A “.maa .-nm.~oA_ aqu _ A E A an _ Ana u 4x . : m.o t\\wu. was a ax . 1 - - .1 1 - u - o.A O .o AA.o u L 11 mA .02 03¢ La.o u c v .fl. A .5 3 . L A; L ® .A. a x quNsoo.o u .N NL: oo~ u a 1/1 apnnildms nuazlno BAIJOIBJ '39“! 140 XL are shown in Fig. 6.24. By investigating these curves, it is observed that the coupled antennas doubly loaded with XL1 = 980 O on the driven antenna and XL2 = 1040 on the parasitic element have uniform current distributions between 0 S 2 S d on both antennas. The input resistance is greatly increased and the input reactance is decreased to a value much smaller than that of unloaded coupled antennas. These physical phenomenon observed are in good agreement with the theoretical prediction for enhanced radiation. This is evidenced by comparing Fig. 6.24 with Fig. 5.6 which shows graphically the corresponding theoretical results developed in Chapter 5. 141 .moccoua< voAaaoo uuvoOA >An=on co mcoAusnAuumAn acouuso AmucuEAuuaxm «N.o .wAm 0A\u doAuAuoa J. 1H1 q 1. 1. 111 . - i o uwh - 0 1 s - - .A / z / I / I :10 i I, 3 / I J! '1' m. / I‘ 8 1 1' } '0' .. "H H J- 1" 1' M 29.5 T o o z .135 #3: / m «0 .m2 AH and 33 z m NA . u 2.8 x H A a 3.3” T 32 8a a a 5.8 I . 05 .m a n boHoaABEN fix 31.x1 / m NAx / W n“ o I a I c o n max 03 x; U c s. . ... m CO¢OHIH1FX HOW III-Ill" (8'8'I'l‘l ill . c 32 a flaw he c 8... u 3x TSN II 1 0 AF 1@ «:2 SN 0 41 $38.0 . 8 T1 is ._ o 1 oLmo.o a L o .1 as .. L 1r1 fl Ifi 1 L NAx NNH [1] [2] [1+] [5] [6] [7] [8] [9] [10] REFERENCES C.W. Harrison, "Monopole with Inductance Loading," IEEE Trans. on Antennas and Propagation, AP-ll, pp. 394-400, July 1961. L. La Paz, and G.A. Miller, "Optimum Current Distributions on Vertical Antennas,” Proc. IRE, 21, pp. 214-232, May 1943. C.J. Bouwkamp, and D.N.G. De Bruijin, "The Problem of Optimum Antenna Current Distribution," Philips Research Reports, 1, pp. 135-158, 1945 H.J. Riblet, "Note on the Maximum Directivity of an Antenna," Proc. IRE, pp. 620-623, May 1948. R.W.P. King, The Theory 2E Linear Antennas, Cambridge, Massachusetts: Harvard University Press, 1956. R.W.P. King, and T.T. Wu, "Currents, Charges, and Near Fields of Cylindrical Antennas," Radio Science Journal of Research NBS/USNC-URSI, 69D, No. 3, pp. 429-446, March 1965. R.W.P. King, Fundamental Electromagnetic Theory, Second Edition, Dover, New York, 1963. E. Hallen, "Theoretical Investigations into the Transmitting and Receiving Qualities of Antennae," Nova Acta Rogiae Soc. Sci. Upsaliensis, 4, No. 2, pp. 1-44, 1938. L.J. Chu, "Physical Limitations of Omi-Directional Antennas," J. Appl. Phys., 12, pp. 1163-1175, December 1948. R.W.P. King, and S.S. Sandler, "The Theory of Broadside Arrays," IEEE Trans. on Antennas and Propagation, AP-12, pp. 269-275, May 1964. 142